Some 3D-Determinant Properties for Calculating of Cubic-Matrix of Order 2 and Order 3

1. Introduction

In this paper we prove some properties for determinant of cubic-matrix of order 2 and order 3. In the paper [2], we have defined the concept of determinant for cubic-matrix of order 2 and order 3, and we have prove some basic properties for calculating this determinants. This idea for developing this concept, it came simply from the determinant of 2D square matrices [19,20,21,24,25,26,29], as well as determinant of rectangular matrices [3,15,16,17,18,30,31,32,33]. In paper [1] we have prove that the Laplace expansion method is valid for calculating the determinant of cubic-matrix for orders 2 and 3. Encouraged by geometric intuition, in this paper we are trying to give an idea and visualize the meaning of the determinants for the cubic-matrix. Our early research mainly lies between geometry, algebra, matrix theory, etc., (see [4,5,6,7,8,9,10,11,12,13,14]).

This paper is continuation of the ideas that arise based on previous researches of 3D matrix ring with element from any whatever field F see [22], but here we study the case when the field F is the field of real numbers $\mathbb{R}$ also is continuation of our research [1,2] related to the study of the properties of determinants for cubic-matrix of order 2 and 3. In this paper we follow a different method from method which is studied in [23].

2. Results for more properties of determinants of cubic-matrix of order 2 and order 3

In this section, all proofs of our theorems are based on the definition of determinant for cubic-matrix of orders 2 and 3, presented in the papers [2] and [1], and results obtained in these papers.

The proofs of the following Theorems are too loaded with indices to calculate, and we are trying to make them a little simpler by separating them case by case, to avoid the difficulty of calculations!

Theorem 1.

Let’s be A and B 3D-cubic matrix with same order (second and third order matrices), then we have that:

$$det(A+B)=det\left(A\right)+det\left(B\right)$$

Proof. 

Case 1. The cubic-matrix A of order 2, (and B has order 2), we will proof the case 1 for each "horizontal layer", "vertical page" and "vertical layer", as following:

1. For plan $i=1$: Let A and B be cubic-matrix of order 2, where all elements on the plan $i=1$ are identical in both matrices, then based on definition of determinant of cubic-matrix presented in [2] and [1] we have:

$$det\left[A_{2\times 2\times 2}\right]+det\left[B_{2\times 2\times 2}\right]=det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)+det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ b_{211}& b_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ b_{212}& b_{222}\end{array}\end{array}\right)$$

$$=a_{111}·a_{222}-a_{112}·a_{221}-a_{121}·a_{212}+a_{122}·a_{211}+a_{111}·b_{222}-a_{112}·b_{221}-a_{121}·b_{212}+a_{122}·b_{211}$$

while,

$$det[A_{2\times 2\times 2}+B_{2\times 2\times 2}]=det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}+b_{211}& a_{221}+b_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ a_{212}+b_{212}& a_{222}+b_{222}\end{array}\end{array}\right)$$

$$=a_{111}·(a_{222}+b_{222})-a_{112}·(a_{221}+b_{221})-a_{121}·(a_{212}+b_{212})+a_{122}·(a_{211}+b_{211})$$

$$=a_{111}·a_{222}+a_{111}·b_{222}-a_{112}·a_{221}-a_{112}·b_{221}-a_{121}·a_{212}-a_{121}·b_{212}+a_{122}·a_{211}+a_{122}·b_{211}$$

If we compare results of above equations, we can see that we have the same result in both cases. Similarly we will proof for all other cases.

2. For plan $i=2$: Let A and B be cubic-matrices of order 2, where all elements on the plan $i=2$ are identical in both matrices, then we have:

$$det\left[A_{2\times 2\times 2}\right]+det\left[B_{2\times 2\times 2}\right]=det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)+det\left(\begin{array}{c}\left.\begin{array}{cc}{b}_{111}& b_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{b}_{112}& b_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)$$

$$=a_{111}·a_{222}-a_{112}·a_{221}-a_{121}·a_{212}+a_{122}·a_{211}+b_{111}·a_{222}-b_{112}·a_{221}-b_{121}·a_{212}+b_{122}·a_{211}$$

while,

$$det[A_{2\times 2\times 2}+B_{2\times 2\times 2}]=det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}+b_{111}& a_{121}+b_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}+b_{112}& a_{122}+b_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)$$

$$=(a_{111}+b_{111})·a_{222}-(a_{112}+b_{112})·a_{221}-(a_{121}+b_{121})·a_{212}+(a_{122}+b_{122})·a_{211}$$

$$=a_{111}·a_{222}+b_{111}·a_{222}-a_{112}·a_{221}-b_{112}·a_{221}-a_{121}·a_{212}-b_{121}·a_{212}+a_{122}·a_{211}+b_{122}·a_{211}$$

If we compare results of above equations, we can see that we have the same result in both cases.

3. For plan $j=1$: Let A and B be cubic-matrices of order 2, where all elements on the plan $j=1$ are identical in both matrices, then we have:

$$det\left[A_{2\times 2\times 2}\right]+det\left[B_{2\times 2\times 2}\right]=det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)+det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& b_{121}\\ a_{211}& b_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& b_{122}\\ a_{212}& b_{222}\end{array}\end{array}\right)$$

$$=a_{111}·a_{222}-a_{112}·a_{221}-a_{121}·a_{212}+a_{122}·a_{211}+a_{111}·b_{222}-a_{112}·b_{221}-b_{121}·a_{212}+b_{122}·a_{211}$$

while,

$$det[A_{2\times 2\times 2}+B_{2\times 2\times 2}]=det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}+b_{121}\\ a_{211}& a_{221}+b_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}+b_{122}\\ a_{212}& a_{222}+b_{222}\end{array}\end{array}\right)$$

$$=a_{111}·(a_{222}+b_{222})-a_{112}·(a_{221}+b_{221})-(a_{121}+b_{121})·a_{212}+(a_{122}+b_{122})·a_{211}$$

$$=a_{111}·a_{222}+a_{111}·b_{222}-a_{112}·a_{221}-a_{112}·b_{221}-a_{121}·a_{212}-b_{121}·a_{212}+a_{122}·a_{211}+b_{122}·a_{211}.$$

If we compare results of above equations, we can see that we have the same result in both cases.

4. For plan $j=2$: Let A and B be cubic-matrices of order 2, where all elements on the plan $j=2$ are identical in both matrices, then we have:

$$det\left[A_{2\times 2\times 2}\right]+det\left[B_{2\times 2\times 2}\right]=det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)+det\left(\begin{array}{c}\left.\begin{array}{cc}{b}_{111}& a_{121}\\ b_{211}& a_{221}\end{array}\right|\begin{array}{cc}{b}_{112}& a_{122}\\ b_{212}& a_{222}\end{array}\end{array}\right)$$

$$=a_{111}·a_{222}-a_{112}·a_{221}-a_{121}·a_{212}+a_{122}·a_{211}+b_{111}·a_{222}-b_{112}·a_{221}-a_{121}·b_{212}+a_{122}·b_{211}$$

while,

$$det[A_{2\times 2\times 2}+B_{2\times 2\times 2}]=det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}+b_{111}& a_{121}\\ a_{211}+b_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}+b_{112}& a_{122}\\ a_{212}+b_{212}& a_{222}\end{array}\end{array}\right)$$

$$=(a_{111}+b_{111})·a_{222}-(a_{112}+b_{112})·a_{221}-a_{121}·(a_{212}+b_{212})+a_{122}·(a_{211}+b_{211})$$

$$=a_{111}·a_{222}+b_{111}·a_{222}-a_{112}·a_{221}-b_{112}·a_{221}-a_{121}·a_{212}-a_{121}·b_{212}+a_{122}·a_{211}+a_{122}·b_{211}$$

If we compare results of above equations, we can see that we have the same result in both cases.

5. For plan $k=1$: Let A and B be cubic-matrices of order 2, where all elements on the plan $k=1$ are identical in both matrices, then we have:

$$det\left[A_{2\times 2\times 2}\right]+det\left[B_{2\times 2\times 2}\right]=det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)+det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{b}_{112}& b_{122}\\ b_{212}& b_{222}\end{array}\end{array}\right)$$

$$=a_{111}·a_{222}-a_{112}·a_{221}-a_{121}·a_{212}+a_{122}·a_{211}+a_{111}·b_{222}-b_{112}·a_{221}-a_{121}·b_{212}+b_{122}·a_{211}$$

while,

$$det[A_{2\times 2\times 2}+B_{2\times 2\times 2}]=det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}+b_{112}& a_{122}+b_{122}\\ a_{212}+b_{212}& a_{222}+b_{222}\end{array}\end{array}\right)$$

$$=a_{111}·(a_{222}+b_{222})-(a_{112}+b_{112})·a_{221}-a_{121}·(a_{212}+b_{212})+(a_{122}+b_{122})·a_{211}$$

$$=a_{111}·a_{222}+a_{111}·b_{222}-a_{112}·a_{221}-b_{112}·a_{221}-a_{121}·a_{212}-a_{121}·b_{212}+a_{122}·a_{211}+b_{122}·a_{211}$$

If we compare results of above equations, we can see that we have the same result in both cases.

6. For plan $k=2$: Let A and B be cubic-matrices of order 2, where all elements on the plan $k=2$ are identical in both matrices, then we have:

$$det\left[A_{2\times 2\times 2}\right]+det\left[B_{2\times 2\times 2}\right]=det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)+det\left(\begin{array}{c}\left.\begin{array}{cc}{b}_{111}& b_{121}\\ b_{211}& b_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)$$

$$=a_{111}·a_{222}-a_{112}·a_{221}-a_{121}·a_{212}+a_{122}·a_{211}+b_{111}·a_{222}-a_{112}·b_{221}-b_{121}·a_{212}+a_{122}·b_{211}$$

while,

$$det[A_{2\times 2\times 2}+B_{2\times 2\times 2}]=det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}+b_{111}& a_{121}+b_{121}\\ a_{211}+b_{211}& a_{221}+b_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)$$

$$=(a_{111}+b_{111})·a_{222}-a_{112}·(a_{221}+b_{221})-(a_{121}+b_{121})·a_{212}+a_{122}·(a_{211}+b_{211})$$

$$=a_{111}·a_{222}+b_{111}·a_{222}-a_{112}·a_{221}-a_{112}·b_{221}-a_{121}·a_{212}-b_{121}·a_{212}+a_{122}·a_{211}+a_{122}·b_{211}$$

If we compare results of above equations, we can see that we have the same result in both cases.

Case 2. The cubic-matrix A of order 3, (and B has order 3), we will proof the case 1 for each "horizontal layer", "vertical page" and "vertical layer", as following:

1. For plan $i=1$: Let A and B be cubic-matrices of order 3, where all elements on the plan $i=1$ and $i=2$ are identical in both matrices, then we have:

$$det\left[A_{3\times 3\times 3}\right]+det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$+det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ b_{311}& b_{321}& b_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ b_{312}& b_{322}& b_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ b_{313}& b_{323}& b_{333}\end{array}\right.\end{array}\right)$$

$$=\{a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}$$

$$+a_{112}·a_{223}·a_{331}+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}$$

$$-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}$$

$$-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}$$

$$-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}+a_{131}·a_{212}·a_{323}$$

$$-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{311}-a_{133}·a_{221}·a_{312}$$

$$+a_{133}·a_{222}·a_{311}\}+\{a_{111}·a_{222}·b_{333}-a_{111}·a_{232}·b_{323}-a_{111}·a_{223}·b_{332}+a_{111}·a_{233}·b_{322}$$

$$-a_{112}·a_{221}·b_{333}+a_{112}·a_{223}·b_{331}+a_{112}·a_{231}·b_{323}-a_{112}·a_{233}·b_{321}+a_{113}·a_{221}·b_{332}$$

$$-a_{113}·a_{222}·b_{331}-a_{113}·a_{231}·b_{322}+a_{113}·a_{232}·b_{321}-a_{121}·a_{212}·b_{333}+a_{121}·a_{213}·b_{332}$$

$$+a_{121}·a_{232}·b_{313}-a_{121}·a_{233}·b_{312}+a_{122}·a_{211}·b_{333}-a_{122}·a_{213}·b_{331}-a_{122}·a_{231}·b_{313}$$

$$+a_{122}·a_{233}·b_{311}-a_{123}·a_{211}·b_{332}+a_{123}·a_{212}·b_{331}+a_{123}·a_{231}·b_{312}-a_{123}·a_{232}·b_{311}$$

$$+a_{131}·a_{212}·b_{323}-a_{131}·a_{213}·b_{322}-a_{131}·a_{222}·b_{313}+a_{131}·a_{223}·b_{312}-a_{132}·a_{211}·b_{323}$$

$$+a_{132}·a_{213}·b_{321}+a_{132}·a_{221}·b_{313}-a_{132}·a_{223}·b_{311}+a_{133}·a_{211}·b_{322}-a_{133}·a_{212}·b_{311}$$

$$-a_{133}·a_{221}·b_{312}+a_{133}·a_{222}·b_{311}\}$$

while,

$$det[A_{3\times 3\times 3}+B_{3\times 3\times 3}]=det\left(\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}+b_{311}& a_{321}+b_{321}& a_{331}+b_{331}\end{array}\right|$$

$$\begin{array}{c}\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}+b_{312}& a_{322}+b_{322}& a_{332}+b_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}+b_{313}& a_{323}+b_{323}& a_{333}+b_{333}\end{array}\right)\end{array}$$

$$=a_{111}·a_{222}·(a_{333}+b_{333})-a_{111}·a_{232}·(a_{323}+b_{323})-a_{111}·a_{223}·(a_{332}+b_{332})+a_{111}·a_{233}·(a_{322}+b_{322})$$

$$-a_{112}·a_{221}·(a_{333}+b_{333})+a_{112}·a_{223}·(a_{331}+b_{331})+a_{112}·a_{231}·(a_{323}+b_{323})-a_{112}·a_{233}·(a_{321}+b_{321})$$

$$+a_{113}·a_{221}·(a_{332}+b_{332})-a_{113}·a_{222}·(a_{331}+b_{331})-a_{113}·a_{231}·(a_{322}+b_{322})+a_{113}·a_{232}·(a_{321}+b_{321})$$

$$-a_{121}·a_{212}·(a_{333}+b_{333})+a_{121}·a_{213}·(a_{332}+b_{332})+a_{121}·a_{232}·(a_{313}+b_{313})-a_{121}·a_{233}·(a_{312}+b_{312})$$

$$+a_{122}·a_{211}·(a_{333}+b_{333})-a_{122}·a_{213}·(a_{331}+b_{331})-a_{122}·a_{231}·(a_{313}+b_{313})+a_{122}·a_{233}·(a_{311}+b_{311})$$

$$-a_{123}·a_{211}·(a_{332}+b_{332})+a_{123}·a_{212}·(a_{331}+b_{331})+a_{123}·a_{231}·(a_{312}+b_{312})-a_{123}·a_{232}·(a_{311}+b_{311})$$

$$+a_{131}·a_{212}·(a_{323}+b_{323})-a_{131}·a_{213}·(a_{322}+b_{322})-a_{131}·a_{222}·(a_{313}+b_{313})+a_{131}·a_{223}·(a_{312}+b_{312})$$

$$-a_{132}·a_{211}·(a_{323}+b_{323})+a_{132}·a_{213}·(a_{321}+b_{321})+a_{132}·a_{221}·(a_{313}+b_{313})-a_{132}·a_{223}·(a_{311}+b_{311})$$

$$+a_{133}·a_{211}·(a_{322}+b_{322})-a_{133}·a_{212}·(a_{311}+b_{311})-a_{133}·a_{221}·(a_{312}+b_{312})+a_{133}·a_{222}·(a_{311}+b_{311}).$$

Hence,

$$det[A_{3\times 3\times 3}+B_{3\times 3\times 3}]=a_{111}·a_{222}·a_{333}+a_{111}·a_{222}·b_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{232}·b_{323}$$

$$-a_{111}·a_{223}·a_{332}-a_{111}·a_{223}·b_{332}+a_{111}·a_{233}·a_{322}+a_{111}·a_{233}·b_{322}-a_{112}·a_{221}·a_{333}$$

$$-a_{112}·a_{221}·b_{333}+a_{112}·a_{223}·a_{331}+a_{112}·a_{223}·b_{331}+a_{112}·a_{231}·a_{323}+a_{112}·a_{231}·b_{323}$$

$$-a_{112}·a_{233}·a_{321}-a_{112}·a_{233}·b_{321}+a_{113}·a_{221}·a_{332}+a_{113}·a_{221}·b_{332}-a_{113}·a_{222}·a_{331}$$

$$-a_{113}·a_{222}·b_{331}-a_{113}·a_{231}·a_{322}-a_{113}·a_{231}·b_{322}+a_{113}·a_{232}·a_{321}+a_{113}·a_{232}·b_{321}$$

$$-a_{121}·a_{212}·a_{333}-a_{121}·a_{212}·b_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{213}·b_{332}+a_{121}·a_{232}·a_{313}$$

$$+a_{121}·a_{232}·b_{313}-a_{121}·a_{233}·a_{312}-a_{121}·a_{233}·b_{312}+a_{122}·a_{211}·a_{333}+a_{122}·a_{211}·b_{333}$$

$$-a_{122}·a_{213}·a_{331}-a_{122}·a_{213}·b_{331}-a_{122}·a_{231}·a_{313}-a_{122}·a_{231}·b_{313}+a_{122}·a_{233}·a_{311}$$

$$+a_{122}·a_{233}·b_{311}-a_{123}·a_{211}·a_{332}-a_{123}·a_{211}·b_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{212}·b_{331}$$

$$+a_{123}·a_{231}·a_{312}+a_{123}·a_{231}·b_{312}-a_{123}·a_{232}·a_{311}-a_{123}·a_{232}·b_{311}+a_{131}·a_{212}·a_{323}$$

$$+a_{131}·a_{212}·b_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{213}·b_{322}-a_{131}·a_{222}·a_{313}-a_{131}·a_{222}·b_{313}$$

$$+a_{131}·a_{223}·a_{312}+a_{131}·a_{223}·b_{312}-a_{132}·a_{211}·a_{323}-a_{132}·a_{211}·b_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{213}·b_{321}+a_{132}·a_{221}·a_{313}+a_{132}·a_{221}·b_{313}-a_{132}·a_{223}·a_{311}-a_{132}·a_{223}·b_{311}$$

$$+a_{133}·a_{211}·a_{322}+a_{133}·a_{211}·b_{322}-a_{133}·a_{212}·a_{311}-a_{133}·a_{212}·b_{311}-a_{133}·a_{221}·a_{312}$$

$$-a_{133}·a_{221}·b_{312}+a_{133}·a_{222}·a_{311}+a_{133}·a_{222}·b_{311}.$$

If we compare results of above equations, we can see that we have the same result in both cases.

2. For plan $i=2$: Let A and B be cubic-matrices of order 3, where all elements on the plan $i=1$ and $i=3$ are identical in both matrices, then we have:

$$det\left[A_{3\times 3\times 3}\right]+det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$+det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ b_{211}& b_{221}& b_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ b_{212}& b_{222}& b_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ b_{213}& b_{223}& b_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=\{a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}$$

$$+a_{112}·a_{223}·a_{331}+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}$$

$$-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}$$

$$-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}$$

$$-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}+a_{131}·a_{212}·a_{323}$$

$$-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{311}-a_{133}·a_{221}·a_{312}$$

$$+a_{133}·a_{222}·a_{311}\}+\{a_{111}·b_{222}·a_{333}-a_{111}·b_{232}·a_{323}-a_{111}·b_{223}·a_{332}+a_{111}·b_{233}·a_{322}$$

$$-a_{112}·b_{221}·a_{333}+a_{112}·b_{223}·a_{331}+a_{112}·b_{231}·a_{323}-a_{112}·b_{233}·a_{321}+a_{113}·b_{221}·a_{332}$$

$$-a_{113}·b_{222}·a_{331}-a_{113}·b_{231}·a_{322}+a_{113}·b_{232}·a_{321}-a_{121}·b_{212}·a_{333}+a_{121}·b_{213}·a_{332}$$

$$+a_{121}·b_{232}·a_{313}-a_{121}·b_{233}·a_{312}+a_{122}·b_{211}·a_{333}-a_{122}·b_{213}·a_{331}-a_{122}·b_{231}·a_{313}$$

$$+a_{122}·b_{233}·a_{311}-a_{123}·b_{211}·a_{332}+a_{123}·b_{212}·a_{331}+a_{123}·b_{231}·a_{312}-a_{123}·b_{232}·a_{311}$$

$$+a_{131}·b_{212}·a_{323}-a_{131}·b_{213}·a_{322}-a_{131}·b_{222}·a_{313}+a_{131}·b_{223}·a_{312}-a_{132}·b_{211}·a_{323}$$

$$+a_{132}·b_{213}·a_{321}+a_{132}·b_{221}·a_{313}-a_{132}·b_{223}·a_{311}+a_{133}·b_{211}·a_{322}-a_{133}·b_{212}·a_{311}$$

$$-a_{133}·b_{221}·a_{312}+a_{133}·b_{222}·a_{311}\}$$

while,

$$det[A_{3\times 3\times 3}+B_{3\times 3\times 3}]=det\left(\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}+b_{211}& a_{221}+b_{221}& a_{231}+b_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|$$

$$\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}+b_{212}& a_{222}+b_{222}& a_{232}+b_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}+b_{213}& a_{223}+b_{223}& a_{233}+b_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right)$$

$$=a_{111}·(a_{222}+b_{222})·a_{333}-a_{111}·(a_{232}+b_{232})·a_{323}-a_{111}·(a_{223}+b_{223})·a_{332}+a_{111}·(a_{233}+b_{233})·a_{322}$$

$$-a_{112}·(a_{221}+b_{221})·a_{333}+a_{112}·(a_{223}+b_{223})·a_{331}+a_{112}·(a_{231}+b_{231})·a_{323}-a_{112}·(a_{233}+b_{233})·a_{321}$$

$$+a_{113}·(a_{221}+b_{221})·a_{332}-a_{113}·(a_{222}+b_{222})·a_{331}-a_{113}·(a_{231}+b_{231})·a_{322}+a_{113}·(a_{232}+b_{232})·a_{321}$$

$$-a_{121}·(a_{212}+b_{212})·a_{333}+a_{121}·(a_{213}+b_{213})·a_{332}+a_{121}·(a_{232}+b_{232})·a_{313}-a_{121}·(a_{233}+b_{233})·a_{312}$$

$$+a_{122}·(a_{211}+b_{211})·a_{333}-a_{122}·(a_{213}+b_{213})·a_{331}-a_{122}·(a_{231}+b_{231})·a_{313}+a_{122}·(a_{233}+b_{233})·a_{311}$$

$$-a_{123}·(a_{211}+b_{211})·a_{332}+a_{123}·(a_{212}+b_{212})·a_{331}+a_{123}·(a_{231}+b_{231})·a_{312}-a_{123}·(a_{232}+b_{232})·a_{311}$$

$$+a_{131}·(a_{212}+b_{212})·a_{323}-a_{131}·(a_{213}+b_{213})·a_{322}-a_{131}·(a_{222}+b_{222})·a_{313}+a_{131}·(a_{223}+b_{223})·a_{312}$$

$$-a_{132}·(a_{211}+b_{211})·a_{323}+a_{132}·(a_{213}+b_{213})·a_{321}+a_{132}·(a_{221}+b_{221})·a_{313}-a_{132}·(a_{223}+b_{223})·a_{311}$$

$$+a_{133}·(a_{211}+b_{211})·a_{322}-a_{133}·(a_{212}+b_{212})·a_{311}-a_{133}·(a_{221}+b_{221})·a_{312}+a_{133}·(a_{222}+b_{222})·a_{311}.$$

Hence,

$$det[A_{3\times 3\times 3}+B_{3\times 3\times 3}]=a_{111}·a_{222}·a_{333}+a_{111}·b_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·b_{232}·a_{323}$$

$$-a_{111}·a_{223}·a_{332}-a_{111}·b_{223}·a_{332}+a_{111}·a_{233}·a_{322}+a_{111}·b_{233}·a_{322}-a_{112}·a_{221}·a_{333}-a_{112}·b_{221}·a_{333}$$

$$+a_{112}·a_{223}·a_{331}+a_{112}·b_{223}·a_{331}+a_{112}·a_{231}·a_{323}+a_{112}·b_{231}·a_{323}-a_{112}·a_{233}·a_{321}-a_{112}·b_{233}·a_{321}$$

$$+a_{113}·a_{221}·a_{332}+a_{113}·b_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·b_{222}·a_{331}-a_{113}·a_{231}·a_{322}-a_{113}·b_{231}·a_{322}$$

$$+a_{113}·a_{232}·a_{321}+a_{113}·b_{232}·a_{321}-a_{121}·a_{212}·a_{333}-a_{121}·b_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·b_{213}·a_{332}$$

$$+a_{121}·a_{232}·a_{313}+a_{121}·b_{232}·a_{313}-a_{121}·a_{233}·a_{312}-a_{121}·b_{233}·a_{312}+a_{122}·a_{211}·a_{333}+a_{122}·b_{211}·a_{333}$$

$$-a_{122}·a_{213}·a_{331}-a_{122}·b_{213}·a_{331}-a_{122}·a_{231}·a_{313}-a_{122}·b_{231}·a_{313}+a_{122}·a_{233}·a_{311}+a_{122}·b_{233}·a_{311}$$

$$-a_{123}·a_{211}·a_{332}-a_{123}·b_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·b_{212}·a_{331}+a_{123}·a_{231}·a_{312}+a_{123}·b_{231}·a_{312}$$

$$-a_{123}·a_{232}·a_{311}-a_{123}·b_{232}·a_{311}+a_{131}·a_{212}·a_{323}+a_{131}·b_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·b_{213}·a_{322}$$

$$-a_{131}·a_{222}·a_{313}-a_{131}·b_{222}·a_{313}+a_{131}·a_{223}·a_{312}+a_{131}·b_{223}·a_{312}-a_{132}·a_{211}·a_{323}-a_{132}·b_{211}·a_{323}$$

$$+a_{132}·a_{213}·a_{321}+a_{132}·b_{213}·a_{321}+a_{132}·a_{221}·a_{313}+a_{132}·b_{221}·a_{313}-a_{132}·a_{223}·a_{311}-a_{132}·b_{223}·a_{311}$$

$$+a_{133}·a_{211}·a_{322}+a_{133}·b_{211}·a_{322}-a_{133}·a_{212}·a_{311}-a_{133}·b_{212}·a_{311}-a_{133}·a_{221}·a_{312}-a_{133}·b_{221}·a_{312}$$

$$+a_{133}·a_{222}·a_{311}+a_{133}·b_{222}·a_{311}$$

If we compare results of above equations, we can see that we have the same result in both cases.

3. For plan $i=3$: Let A and B be cubic-matrices of order 3, where all elements on the plan $i=2$ and $i=3$ are identical in both matrices, then we have:

$$det\left[A_{3\times 3\times 3}\right]+det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$+det\left(\begin{array}{c}\left.\begin{array}{ccc}{b}_{111}& b_{121}& b_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{b}_{112}& b_{122}& b_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{b}_{113}& b_{123}& b_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=\{a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{311}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}\}$$

$$+\{b_{111}·a_{222}·a_{333}-b_{111}·a_{232}·a_{323}-b_{111}·a_{223}·a_{332}+b_{111}·a_{233}·a_{322}-b_{112}·a_{221}·a_{333}+b_{112}·a_{223}·a_{331}$$

$$+b_{112}·a_{231}·a_{323}-b_{112}·a_{233}·a_{321}+b_{113}·a_{221}·a_{332}-b_{113}·a_{222}·a_{331}-b_{113}·a_{231}·a_{322}+b_{113}·a_{232}·a_{321}$$

$$-b_{121}·a_{212}·a_{333}+b_{121}·a_{213}·a_{332}+b_{121}·a_{232}·a_{313}-b_{121}·a_{233}·a_{312}+b_{122}·a_{211}·a_{333}-b_{122}·a_{213}·a_{331}$$

$$-b_{122}·a_{231}·a_{313}+b_{122}·a_{233}·a_{311}-b_{123}·a_{211}·a_{332}+b_{123}·a_{212}·a_{331}+b_{123}·a_{231}·a_{312}-b_{123}·a_{232}·a_{311}$$

$$+b_{131}·a_{212}·a_{323}-b_{131}·a_{213}·a_{322}-b_{131}·a_{222}·a_{313}+b_{131}·a_{223}·a_{312}-b_{132}·a_{211}·a_{323}+b_{132}·a_{213}·a_{321}$$

$$+b_{132}·a_{221}·a_{313}-b_{132}·a_{223}·a_{311}+b_{133}·a_{211}·a_{322}-b_{133}·a_{212}·a_{311}-b_{133}·a_{221}·a_{312}+b_{133}·a_{222}·a_{311}\}$$

while,

$$det[A_{3\times 3\times 3}+B_{3\times 3\times 3}]=det\left(\begin{array}{ccc}{a}_{111}+b_{111}& a_{121}+b_{121}& a_{131}+b_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|$$

$$\begin{array}{ccc}{a}_{112}+b_{112}& a_{122}+b_{122}& a_{132}+b_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}+b_{113}& a_{123}+b_{123}& a_{133}+b_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right)$$

$$=(a_{111}+b_{111})·a_{222}·a_{333}-(a_{111}+b_{111})·a_{232}·a_{323}-(a_{111}+b_{111})·a_{223}·a_{332}+(a_{111}+b_{111})·a_{233}·a_{322}$$

$$-(a_{112}+b_{112})·a_{221}·a_{333}+(a_{112}+b_{112})·a_{223}·a_{331}+(a_{112}+b_{112})·a_{231}·a_{323}-(a_{112}+b_{112})·a_{233}·a_{321}$$

$$+(a_{113}+b_{113})·a_{221}·a_{332}-(a_{113}+b_{113})·a_{222}·a_{331}-(a_{113}+b_{113})·a_{231}·a_{322}+(a_{113}+b_{113})·a_{232}·a_{321}$$

$$-(a_{121}+b_{121})·a_{212}·a_{333}+(a_{121}+b_{121})·a_{213}·a_{332}+(a_{121}+b_{121})·a_{232}·a_{313}-(a_{121}+b_{121})·a_{233}·a_{312}$$

$$+(a_{122}+b_{122})·a_{211}·a_{333}-(a_{122}+b_{122})·a_{213}·a_{331}-(a_{122}+b_{122})·a_{231}·a_{313}+(a_{122}+b_{122})·a_{233}·a_{311}$$

$$-(a_{123}+b_{123})·a_{211}·a_{332}+(a_{123}+b_{123})·a_{212}·a_{331}+(a_{123}+b_{123})·a_{231}·a_{312}-(a_{123}+b_{123})·a_{232}·a_{311}$$

$$+(a_{131}+b_{131})·a_{212}·a_{323}-(a_{131}+b_{131})·a_{213}·a_{322}-(a_{131}+b_{131})·a_{222}·a_{313}+(a_{131}+b_{131})·a_{223}·a_{312}$$

$$-(a_{132}+b_{132})·a_{211}·a_{323}+(a_{132}+b_{132})·a_{213}·a_{321}+(a_{132}+b_{132})·a_{221}·a_{313}-(a_{132}+b_{132})·a_{223}·a_{311}$$

$$+(a_{133}+b_{133})·a_{211}·a_{322}-(a_{133}+b_{133})·a_{212}·a_{311}-(a_{133}+b_{133})·a_{221}·a_{312}+(a_{133}+b_{133})·a_{222}·a_{311}$$

Hence,

$$det\left[A_{3\times 3\times 3}\right]+B_{3\times 3\times 3}]=a_{111}·a_{222}·a_{333}+b_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-b_{111}·a_{232}·a_{323}$$

$$-a_{111}·a_{223}·a_{332}-b_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}+b_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}-b_{112}·a_{221}·a_{333}$$

$$+a_{112}·a_{223}·a_{331}+b_{112}·a_{223}·a_{331}+a_{112}·a_{231}·a_{323}+b_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}-b_{112}·a_{233}·a_{321}$$

$$+a_{113}·a_{221}·a_{332}+b_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-b_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}-b_{113}·a_{231}·a_{322}$$

$$+a_{113}·a_{232}·a_{321}+b_{113}·a_{232}·a_{321}-a_{121}·a_{212}·a_{333}-b_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+b_{121}·a_{213}·a_{332}$$

$$+a_{121}·a_{232}·a_{313}+b_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}-b_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}+b_{122}·a_{211}·a_{333}$$

$$-a_{122}·a_{213}·a_{331}-b_{122}·a_{213}·a_{331}-a_{122}·a_{231}·a_{313}-b_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}+b_{122}·a_{233}·a_{311}$$

$$-a_{123}·a_{211}·a_{332}-b_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+b_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}+b_{123}·a_{231}·a_{312}$$

$$-a_{123}·a_{232}·a_{311}-b_{123}·a_{232}·a_{311}+a_{131}·a_{212}·a_{323}+b_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-b_{131}·a_{213}·a_{322}$$

$$-a_{131}·a_{222}·a_{313}-b_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}+b_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}-b_{132}·a_{211}·a_{323}$$

$$+a_{132}·a_{213}·a_{321}+b_{132}·a_{213}·a_{321}+a_{132}·a_{221}·a_{313}+b_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}-b_{132}·a_{223}·a_{311}$$

$$+a_{133}·a_{211}·a_{322}+b_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{311}-b_{133}·a_{212}·a_{311}-a_{133}·a_{221}·a_{312}-b_{133}·a_{221}·a_{312}$$

$$+a_{133}·a_{222}·a_{311}+b_{133}·a_{222}·a_{311}.$$

If we compare results of above equations, we can see that we have the same result in both cases.

4. For plan $j=1$: Let A and B be cubic-matrices of order 3, where all elements on the plan $j=1$ and $j=2$ are identical in both matrices, then have:

$$det\left[A_{3\times 3\times 3}\right]+det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$+det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& b_{131}\\ a_{211}& a_{221}& b_{231}\\ a_{311}& a_{321}& b_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& b_{132}\\ a_{212}& a_{222}& b_{232}\\ a_{312}& a_{322}& b_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& b_{133}\\ a_{213}& a_{223}& b_{233}\\ a_{313}& a_{323}& b_{333}\end{array}\right.\end{array}\right)$$

$$=\{a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{311}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}\}$$

$$+\{a_{111}·a_{222}·b_{333}-a_{111}·b_{232}·a_{323}-a_{111}·a_{223}·b_{332}+a_{111}·b_{233}·a_{322}-a_{112}·a_{221}·b_{333}+a_{112}·a_{223}·b_{331}$$

$$+a_{112}·b_{231}·a_{323}-a_{112}·b_{233}·a_{321}+a_{113}·a_{221}·b_{332}-a_{113}·a_{222}·b_{331}-a_{113}·b_{231}·a_{322}+a_{113}·b_{232}·a_{321}$$

$$-a_{121}·a_{212}·b_{333}+a_{121}·a_{213}·b_{332}+a_{121}·b_{232}·a_{313}-a_{121}·b_{233}·a_{312}+a_{122}·a_{211}·b_{333}-a_{122}·a_{213}·b_{331}$$

$$-a_{122}·b_{231}·a_{313}+a_{122}·b_{233}·a_{311}-a_{123}·a_{211}·b_{332}+a_{123}·a_{212}·b_{331}+a_{123}·b_{231}·a_{312}-a_{123}·b_{232}·a_{311}$$

$$+b_{131}·a_{212}·a_{323}-b_{131}·a_{213}·a_{322}-b_{131}·a_{222}·a_{313}+b_{131}·a_{223}·a_{312}-b_{132}·a_{211}·a_{323}+b_{132}·a_{213}·a_{321}$$

$$+b_{132}·a_{221}·a_{313}-b_{132}·a_{223}·a_{311}+b_{133}·a_{211}·a_{322}-b_{133}·a_{212}·a_{311}-b_{133}·a_{221}·a_{312}+b_{133}·a_{222}·a_{311}\}$$

while,

$$det[A_{3\times 3\times 3}+B_{3\times 3\times 3}]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}+b_{131}\\ a_{211}& a_{221}& a_{231}+b_{231}\\ a_{311}& a_{321}& a_{331}+b_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}+b_{132}\\ a_{212}& a_{222}& a_{232}+b_{232}\\ a_{312}& a_{322}& a_{332}+b_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}+b_{133}\\ a_{213}& a_{223}& a_{233}+b_{233}\\ a_{313}& a_{323}& a_{333}+b_{333}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·(a_{333}+b_{333})-a_{111}·(a_{232}+b_{232})·a_{323}-a_{111}·a_{223}·(a_{332}+b_{332})+a_{111}·(a_{233}+b_{233})·a_{322}$$

$$-a_{112}·a_{221}·(a_{333}+b_{333})+a_{112}·a_{223}·(a_{331}+b_{331})+a_{112}·(a_{231}+b_{231})·a_{323}-a_{112}·(a_{233}+b_{233})·a_{321}$$

$$+a_{113}·a_{221}·(a_{332}+b_{332})-a_{113}·a_{222}·(a_{331}+b_{331})-a_{113}·(a_{231}+b_{231})·a_{322}+a_{113}·(a_{232}+b_{232})·a_{321}$$

$$-a_{121}·a_{212}·(a_{333}+b_{333})+a_{121}·a_{213}·(a_{332}+b_{332})+a_{121}·(a_{232}+b_{232})·a_{313}-a_{121}·(a_{233}+b_{233})·a_{312}$$

$$+a_{122}·a_{211}·(a_{333}+b_{333})-a_{122}·a_{213}·(a_{331}+b_{331})-a_{122}·(a_{231}+b_{231})·a_{313}+a_{122}·(a_{233}+b_{233})·a_{311}$$

$$-a_{123}·a_{211}·(a_{332}+b_{332})+a_{123}·a_{212}·(a_{331}+b_{331})+a_{123}·(a_{231}+b_{231})·a_{312}-a_{123}·(a_{232}+b_{232})·a_{311}$$

$$+(a_{131}+b_{131})·a_{212}·a_{323}-(a_{131}+b_{131})·a_{213}·a_{322}-(a_{131}+b_{131})·a_{222}·a_{313}+(a_{131}+b_{131})·a_{223}·a_{312}$$

$$-(a_{132}+b_{132})·a_{211}·a_{323}+(a_{132}+b_{132})·a_{213}·a_{321}+(a_{132}+b_{132})·a_{221}·a_{313}-(a_{132}+b_{132})·a_{223}·a_{311}$$

$$+(a_{133}+b_{133})·a_{211}·a_{322}-(a_{133}+b_{133})·a_{212}·a_{311}-(a_{133}+b_{133})·a_{221}·a_{312}+(a_{133}+b_{133})·a_{222}·a_{311}$$

$$=a_{111}·a_{222}·a_{333}+a_{111}·a_{222}·b_{333}-a_{111}·a_{232}·a_{323}-a_{111}·b_{232}·a_{323}-a_{111}·a_{223}·a_{332}-a_{111}·a_{223}·b_{332}$$

$$+a_{111}·a_{233}·a_{322}+a_{111}·b_{233}·a_{322}-a_{112}·a_{221}·a_{333}-a_{112}·a_{221}·b_{333}+a_{112}·a_{223}·a_{331}+a_{112}·b_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}+a_{112}·b_{231}·a_{323}-a_{112}·a_{233}·a_{321}-a_{112}·b_{233}·a_{321}+a_{113}·a_{221}·a_{332}+a_{113}·a_{221}·b_{332}$$

$$-a_{113}·a_{222}·a_{331}-a_{113}·a_{222}·b_{331}-a_{113}·a_{231}·a_{322}-a_{113}·b_{231}·a_{322}+a_{113}·a_{232}·a_{321}+a_{113}·b_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}-a_{121}·a_{212}·b_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{213}·b_{332}+a_{121}·a_{232}·a_{313}+a_{121}·b_{232}·a_{313}$$

$$-a_{121}·a_{233}·a_{312}-a_{121}·b_{233}·a_{312}+a_{122}·a_{211}·a_{333}+a_{122}·a_{211}·b_{333}-a_{122}·a_{213}·a_{331}-a_{122}·a_{213}·b_{331}$$

$$-a_{122}·a_{231}·a_{313}-a_{122}·b_{231}·a_{313}+a_{122}·a_{233}·a_{311}+a_{122}·b_{233}·a_{311}-a_{123}·a_{211}·a_{332}-a_{123}·a_{211}·b_{332}$$

$$+a_{123}·a_{212}·a_{331}+a_{123}·a_{212}·b_{331}+a_{123}·a_{231}·a_{312}+a_{123}·b_{231}·a_{312}-a_{123}·a_{232}·a_{311}-a_{123}·b_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}+b_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-b_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}-b_{131}·a_{222}·a_{313}$$

$$+a_{131}·a_{223}·a_{312}+b_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}-b_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}+b_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}+b_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}-b_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}+b_{133}·a_{211}·a_{322}$$

$$-a_{133}·a_{212}·a_{311}-b_{133}·a_{212}·a_{311}-a_{133}·a_{221}·a_{312}-b_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}+b_{133}·a_{222}·a_{311}$$

If we compare results of above equations, we can see that we have the same result in both cases.

5. For plan $j=2$: Let A and B be cubic-matrices of order 3, where all elements on the plan $j=1$ and $j=3$ are identical in both matrices, then have:

$$det\left[A_{3\times 3\times 3}\right]+det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$+det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& b_{121}& a_{131}\\ a_{211}& b_{221}& a_{231}\\ a_{311}& b_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& b_{122}& a_{132}\\ a_{212}& b_{222}& a_{232}\\ a_{312}& b_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& b_{123}& a_{133}\\ a_{213}& b_{223}& a_{233}\\ a_{313}& b_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=\{a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}\}$$

$$+\{a_{111}·b_{222}·a_{333}-a_{111}·a_{232}·b_{323}-a_{111}·b_{223}·a_{332}+a_{111}·a_{233}·b_{322}-a_{112}·b_{221}·a_{333}+a_{112}·b_{223}·a_{331}$$

$$+a_{112}·a_{231}·b_{323}-a_{112}·a_{233}·b_{321}+a_{113}·b_{221}·a_{332}-a_{113}·b_{222}·a_{331}-a_{113}·a_{231}·b_{322}+a_{113}·a_{232}·b_{321}$$

$$-b_{121}·a_{212}·a_{333}+b_{121}·a_{213}·a_{332}+b_{121}·a_{232}·a_{313}-b_{121}·a_{233}·a_{312}+b_{122}·a_{211}·a_{333}-b_{122}·a_{213}·a_{331}$$

$$-b_{122}·a_{231}·a_{313}+b_{122}·a_{233}·a_{311}-b_{123}·a_{211}·a_{332}+b_{123}·a_{212}·a_{331}+b_{123}·a_{231}·a_{312}-b_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·b_{323}-a_{131}·a_{213}·b_{322}-a_{131}·b_{222}·a_{313}+a_{131}·b_{223}·a_{312}-a_{132}·a_{211}·b_{323}+a_{132}·a_{213}·b_{321}$$

$$+a_{132}·b_{221}·a_{313}-a_{132}·b_{223}·a_{311}+a_{133}·a_{211}·b_{322}-a_{133}·a_{212}·a_{321}-a_{133}·b_{221}·a_{312}+a_{133}·b_{222}·a_{311}\}$$

while,

$$det[A_{3\times 3\times 3}+B_{3\times 3\times 3}]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}+b_{121}& a_{131}\\ a_{211}& a_{221}+b_{221}& a_{231}\\ a_{311}& a_{321}+b_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}+b_{122}& a_{132}\\ a_{212}& a_{222}+b_{222}& a_{232}\\ a_{312}& a_{322}+b_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}+b_{123}& a_{133}\\ a_{213}& a_{223}+b_{223}& a_{233}\\ a_{313}& a_{323}+b_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=a_{111}·(a_{222}+b_{222})·a_{333}-a_{111}·a_{232}·(a_{323}+b_{323})-a_{111}·(a_{223}+b_{223})·a_{332}+a_{111}·a_{233}·(a_{322}+b_{322})$$

$$-a_{112}·(a_{221}+b_{221})·a_{333}+a_{112}·(a_{223}+b_{223})·a_{331}+a_{112}·a_{231}·(a_{323}+b_{323})-a_{112}·a_{233}·(a_{321}+b_{321})$$

$$+a_{113}·(a_{221}+b_{221})·a_{332}-a_{113}·(a_{222}+b_{222})·a_{331}-a_{113}·a_{231}·(a_{322}+b_{322})+a_{113}·a_{232}·(a_{321}+b_{321})$$

$$-(a_{121}+b_{121})·a_{212}·a_{333}+(a_{121}+b_{121})·a_{213}·a_{332}+(a_{121}+b_{121})·a_{232}·a_{313}-(a_{121}+b_{121})·a_{233}·a_{312}$$

$$+(a_{122}+b_{122})·a_{211}·a_{333}-(a_{122}+b_{122})·a_{213}·a_{331}-(a_{122}+b_{122})·a_{231}·a_{313}+(a_{122}+b_{122})·a_{233}·a_{311}$$

$$-(a_{123}+b_{123})·a_{211}·a_{332}+(a_{123}+b_{123})·a_{212}·a_{331}+(a_{123}+b_{123})·a_{231}·a_{312}-(a_{123}+b_{123})·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·(a_{323}+b_{323})-a_{131}·a_{213}·(a_{322}+b_{322})-a_{131}·(a_{222}+b_{222})·a_{313}+a_{131}·(a_{223}+b_{223})·a_{312}$$

$$-a_{132}·a_{211}·(a_{323}+b_{323})+a_{132}·a_{213}·(a_{321}+b_{321})+a_{132}·(a_{221}+b_{221})·a_{313}-a_{132}·(a_{223}+b_{223})·a_{311}$$

$$+a_{133}·a_{211}·(a_{322}+b_{322})-a_{133}·a_{212}·(a_{321}+b_{321})-a_{133}·(a_{221}+b_{221})·a_{312}+a_{133}·(a_{222}+b_{222})·a_{311}$$

$$=a_{111}·a_{222}·a_{333}+a_{111}·b_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{232}·b_{323}-a_{111}·a_{223}·a_{332}-a_{111}·b_{223}·a_{332}$$

$$+a_{111}·a_{233}·a_{322}+a_{111}·a_{233}·b_{322}-a_{112}·a_{221}·a_{333}-a_{112}·b_{221}·a_{333}+a_{112}·a_{223}·a_{331}+a_{112}·b_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}+a_{112}·a_{231}·b_{323}-a_{112}·a_{233}·a_{321}-a_{112}·a_{233}·b_{321}+a_{113}·a_{221}·a_{332}+a_{113}·b_{221}·a_{332}$$

$$-a_{113}·a_{222}·a_{331}-a_{113}·b_{222}·a_{331}-a_{113}·a_{231}·a_{322}-a_{113}·a_{231}·b_{322}+a_{113}·a_{232}·a_{321}+a_{113}·a_{232}·b_{321}$$

$$-a_{121}·a_{212}·a_{333}-b_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+b_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}+b_{121}·a_{232}·a_{313}$$

$$-a_{121}·a_{233}·a_{312}-b_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}+b_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}-b_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}-b_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}+b_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}-b_{123}·a_{211}·a_{332}$$

$$+a_{123}·a_{212}·a_{331}+b_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}+b_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}-b_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}+a_{131}·a_{212}·b_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{213}·b_{322}-a_{131}·a_{222}·a_{313}-a_{131}·b_{222}·a_{313}$$

$$+a_{131}·a_{223}·a_{312}+a_{131}·b_{223}·a_{312}-a_{132}·a_{211}·a_{323}-a_{132}·a_{211}·b_{323}+a_{132}·a_{213}·a_{321}+a_{132}·a_{213}·b_{321}$$

$$+a_{132}·a_{221}·a_{313}+a_{132}·b_{221}·a_{313}-a_{132}·a_{223}·a_{311}-a_{132}·b_{223}·a_{311}+a_{133}·a_{211}·a_{322}+a_{133}·a_{211}·b_{322}$$

$$-a_{133}·a_{212}·a_{321}-a_{133}·a_{212}·b_{321}-a_{133}·a_{221}·a_{312}-a_{133}·b_{221}·a_{312}+a_{133}·a_{222}·a_{311}+a_{133}·b_{222}·a_{311}$$

If we compare results of above equations, we can see that we have the same result in both cases.

6. For plan $j=3$: Let A and B be cubic-matrices of order 3, where all elements on the plan $j=3$ and $j=3$ are identical in both matrices, then have:

$$det\left[A_{3\times 3\times 3}\right]+det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$+det\left(\begin{array}{c}\left.\begin{array}{ccc}{b}_{111}& a_{121}& a_{131}\\ b_{211}& a_{221}& a_{231}\\ b_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{b}_{112}& a_{122}& a_{132}\\ b_{212}& a_{222}& a_{232}\\ b_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{b}_{113}& a_{123}& a_{133}\\ b_{213}& a_{223}& a_{233}\\ b_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=\{a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}\}$$

$$+\{b_{111}·a_{222}·a_{333}-b_{111}·a_{232}·a_{323}-b_{111}·a_{223}·a_{332}+b_{111}·a_{233}·a_{322}-b_{112}·a_{221}·a_{333}+b_{112}·a_{223}·a_{331}$$

$$+b_{112}·a_{231}·a_{323}-b_{112}·a_{233}·a_{321}+b_{113}·a_{221}·a_{332}-b_{113}·a_{222}·a_{331}-b_{113}·a_{231}·a_{322}+b_{113}·a_{232}·a_{321}$$

$$-a_{121}·b_{212}·a_{333}+a_{121}·b_{213}·a_{332}+a_{121}·a_{232}·b_{313}-a_{121}·a_{233}·b_{312}+a_{122}·b_{211}·a_{333}-a_{122}·b_{213}·a_{331}$$

$$-a_{122}·a_{231}·b_{313}+a_{122}·a_{233}·b_{311}-a_{123}·b_{211}·a_{332}+a_{123}·b_{212}·a_{331}+a_{123}·a_{231}·b_{312}-a_{123}·a_{232}·b_{311}$$

$$+a_{131}·b_{212}·a_{323}-a_{131}·b_{213}·a_{322}-a_{131}·a_{222}·b_{313}+a_{131}·a_{223}·b_{312}-a_{132}·b_{211}·a_{323}+a_{132}·b_{213}·a_{321}$$

$$+a_{132}·a_{221}·b_{313}-a_{132}·a_{223}·b_{311}+a_{133}·b_{211}·a_{322}-a_{133}·b_{212}·a_{321}-a_{133}·a_{221}·b_{312}+a_{133}·a_{222}·b_{311}\}$$

while,

$$det[A_{3\times 3\times 3}+B_{3\times 3\times 3}]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}+b_{111}& a_{121}& a_{131}\\ a_{211}+b_{211}& a_{221}& a_{231}\\ a_{311}+b_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}+b_{112}& a_{122}& a_{132}\\ a_{212}+b_{212}& a_{222}& a_{232}\\ a_{312}+b_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}+b_{113}& a_{123}& a_{133}\\ a_{213}+b_{213}& a_{223}& a_{233}\\ a_{131}+b_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=(a_{111}+b_{111})·a_{222}·a_{333}-(a_{111}+b_{111})·a_{232}·a_{323}-(a_{111}+b_{111})·a_{223}·a_{332}+(a_{111}+b_{111})·a_{233}·a_{322}$$

$$-(a_{112}+b_{112})·a_{221}·a_{333}+(a_{112}+b_{112})·a_{223}·a_{331}+(a_{112}+b_{112})·a_{231}·a_{323}-(a_{112}+b_{112})·a_{233}·a_{321}$$

$$+(a_{113}+b_{113})·a_{221}·a_{332}-(a_{113}+b_{113})·a_{222}·a_{331}-(a_{113}+b_{113})·a_{231}·a_{322}+(a_{113}+b_{113})·a_{232}·a_{321}$$

$$-a_{121}·(a_{212}+b_{212})·a_{333}+a_{121}·(a_{213}+b_{213})·a_{332}+a_{121}·a_{232}·(a_{313}+b_{313})-a_{121}·a_{233}·(a_{312}+b_{312})$$

$$+a_{122}·(a_{211}+b_{211})·a_{333}-a_{122}·(a_{213}+b_{213})·a_{331}-a_{122}·a_{231}·(a_{313}+b_{313})+a_{122}·a_{233}·(a_{311}+b_{311})$$

$$-a_{123}·(a_{211}+b_{211})·a_{332}+a_{123}·(a_{212}+b_{212})·a_{331}+a_{123}·a_{231}·(a_{312}+b_{312})-a_{123}·a_{232}·(a_{311}+b_{311})$$

$$+a_{131}·(a_{212}+b_{212})·a_{323}-a_{131}·(a_{213}+b_{213})·a_{322}-a_{131}·a_{222}·(a_{313}+b_{313})+a_{131}·a_{223}·(a_{312}+b_{312})$$

$$-a_{132}·(a_{211}+b_{211})·a_{323}+a_{132}·(a_{213}+b_{213})·a_{321}+a_{132}·a_{221}·(a_{313}+b_{313})-a_{132}·a_{223}·(a_{311}+b_{311})$$

$$+a_{133}·(a_{211}+b_{211})·a_{322}-a_{133}·(a_{212}+b_{212})·a_{321}-a_{133}·a_{221}·(a_{312}+b_{312})+a_{133}·a_{222}·(a_{311}+b_{311})$$

$$=a_{111}·a_{222}·a_{333}+b_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-b_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}-b_{111}·a_{223}·a_{332}$$

$$+a_{111}·a_{233}·a_{322}+b_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}-b_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}+b_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}+b_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}-b_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}+b_{113}·a_{221}·a_{332}$$

$$-a_{113}·a_{222}·a_{331}-b_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}-b_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}+b_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}-a_{121}·b_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·b_{213}·a_{332}+a_{121}·a_{232}·a_{313}+a_{121}·a_{232}·b_{313}$$

$$-a_{121}·a_{233}·a_{312}-a_{121}·a_{233}·b_{312}+a_{122}·a_{211}·a_{333}+a_{122}·b_{211}·a_{333}-a_{122}·a_{213}·a_{331}-a_{122}·b_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}-a_{122}·a_{231}·b_{313}+a_{122}·a_{233}·a_{311}+a_{122}·a_{233}·b_{311}-a_{123}·a_{211}·a_{332}-a_{123}·b_{211}·a_{332}$$

$$+a_{123}·a_{212}·a_{331}+a_{123}·b_{212}·a_{331}+a_{123}·a_{231}·a_{312}+a_{123}·a_{231}·b_{312}-a_{123}·a_{232}·a_{311}-a_{123}·a_{232}·b_{311}$$

$$+a_{131}·a_{212}·a_{323}+a_{131}·b_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·b_{213}·a_{322}-a_{131}·a_{222}·a_{313}-a_{131}·a_{222}·b_{313}$$

$$+a_{131}·a_{223}·a_{312}+a_{131}·a_{223}·b_{312}-a_{132}·a_{211}·a_{323}-a_{132}·b_{211}·a_{323}+a_{132}·a_{213}·a_{321}+a_{132}·b_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}+a_{132}·a_{221}·b_{313}-a_{132}·a_{223}·a_{311}-a_{132}·a_{223}·b_{311}+a_{133}·a_{211}·a_{322}+a_{133}·b_{211}·a_{322}$$

$$-a_{133}·a_{212}·a_{321}-a_{133}·b_{212}·a_{321}-a_{133}·a_{221}·a_{312}-a_{133}·a_{221}·b_{312}+a_{133}·a_{222}·a_{311}+a_{133}·a_{222}·b_{311}$$

If we compare results of above equations, we can see that we have the same result in both cases.

7. For plan $k=1$: Let A and B be cubic-matrices of order 3, where all elements on the plan $k=1$ and $k=2$ are identical in both matrices, then have:

$$det\left[A_{3\times 3\times 3}\right]+det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$+det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{b}_{113}& b_{123}& b_{133}\\ b_{213}& b_{223}& b_{233}\\ b_{313}& b_{323}& b_{333}\end{array}\right.\end{array}\right)$$

$$=\{a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}\}$$

$$+\{a_{111}·a_{222}·b_{333}-a_{111}·a_{232}·b_{323}-a_{111}·b_{223}·a_{332}+a_{111}·b_{233}·a_{322}-a_{112}·a_{221}·b_{333}+a_{112}·b_{223}·a_{331}$$

$$+a_{112}·a_{231}·b_{323}-a_{112}·b_{233}·a_{321}+b_{113}·a_{221}·a_{332}-b_{113}·a_{222}·a_{331}-b_{113}·a_{231}·a_{322}+b_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·b_{333}+a_{121}·b_{213}·a_{332}+a_{121}·a_{232}·b_{313}-a_{121}·b_{233}·a_{312}+a_{122}·a_{211}·b_{333}-a_{122}·b_{213}·a_{331}$$

$$-a_{122}·a_{231}·b_{313}+a_{122}·b_{233}·a_{311}-b_{123}·a_{211}·a_{332}+b_{123}·a_{212}·a_{331}+b_{123}·a_{231}·a_{312}-b_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·b_{323}-a_{131}·b_{213}·a_{322}-a_{131}·a_{222}·b_{313}+a_{131}·b_{223}·a_{312}-a_{132}·a_{211}·b_{323}+a_{132}·b_{213}·a_{321}$$

$$+a_{132}·a_{221}·b_{313}-a_{132}·b_{223}·a_{311}+b_{133}·a_{211}·a_{322}-b_{133}·a_{212}·a_{321}-b_{133}·a_{221}·a_{312}+b_{133}·a_{222}·a_{311}\}$$

while,

$$det[A_{3\times 3\times 3}+B_{3\times 3\times 3}]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}+b_{113}& a_{123}+b_{123}& a_{133}+b_{133}\\ a_{213}+b_{213}& a_{223}+b_{223}& a_{233}+b_{233}\\ a_{313}+b_{313}& a_{323}+b_{323}& a_{333}+b_{333}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·(a_{333}+b_{333})-a_{111}·a_{232}·(a_{323}+b_{323})-a_{111}·(a_{223}+b_{223})·a_{332}+a_{111}·(a_{233}+b_{233})·a_{322}$$

$$-a_{112}·a_{221}·(a_{333}+b_{333})+a_{112}·(a_{223}+b_{223})·a_{331}+a_{112}·a_{231}·(a_{323}+b_{323})-a_{112}·(a_{233}+b_{233})·a_{321}$$

$$+(a_{113}+b_{113})·a_{221}·a_{332}-(a_{113}+b_{113})·a_{222}·a_{331}-(a_{113}+b_{113})·a_{231}·a_{322}+(a_{113}+b_{113})·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·(a_{333}+b_{333})+a_{121}·(a_{213}+b_{213})·a_{332}+a_{121}·a_{232}·(a_{313}+b_{313})-a_{121}·(a_{233}+b_{233})·a_{312}$$

$$+a_{122}·a_{211}·(a_{333}+b_{333})-a_{122}·(a_{213}+b_{213})·a_{331}-a_{122}·a_{231}·(a_{313}+b_{313})+a_{122}·(a_{233}+b_{233})·a_{311}$$

$$-(a_{123}+b_{123})·a_{211}·a_{332}+(a_{123}+b_{123})·a_{212}·a_{331}+(a_{123}+b_{123})·a_{231}·a_{312}-(a_{123}+b_{123})·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·(a_{323}+b_{323})-a_{131}·(a_{213}+b_{213})·a_{322}-a_{131}·a_{222}·(a_{313}+b_{313})+a_{131}·(a_{223}+b_{223})·a_{312}$$

$$-a_{132}·a_{211}·(a_{323}+b_{323})+a_{132}·(a_{213}+b_{213})·a_{321}+a_{132}·a_{221}·(a_{313}+b_{313})-a_{132}·(a_{223}+b_{223})·a_{311}$$

$$+(a_{133}+b_{133})·a_{211}·a_{322}-(a_{133}+b_{133})·a_{212}·a_{321}-(a_{133}+b_{133})·a_{221}·a_{312}+(a_{133}+b_{133})·a_{222}·a_{311}$$

$$=a_{111}·a_{222}·a_{333}+a_{111}·a_{222}·b_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{232}·b_{323}-a_{111}·a_{223}·a_{332}-a_{111}·b_{223}·a_{332}$$

$$+a_{111}·a_{233}·a_{322}+a_{111}·b_{233}·a_{322}-a_{112}·a_{221}·a_{333}-a_{112}·a_{221}·b_{333}+a_{112}·a_{223}·a_{331}+a_{112}·b_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}+a_{112}·a_{231}·b_{323}-a_{112}·a_{233}·a_{321}-a_{112}·b_{233}·a_{321}+a_{113}·a_{221}·a_{332}+b_{113}·a_{221}·a_{332}$$

$$-a_{113}·a_{222}·a_{331}-b_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}-b_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}+b_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}-a_{121}·a_{212}·b_{333}+a_{121}·a_{213}·a_{332}+a_{121}·b_{213}·a_{332}+a_{121}·a_{232}·a_{313}+a_{121}·a_{232}·b_{313}$$

$$-a_{121}·a_{233}·a_{312}-a_{121}·b_{233}·a_{312}+a_{122}·a_{211}·a_{333}+a_{122}·a_{211}·b_{333}-a_{122}·a_{213}·a_{331}-a_{122}·b_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}-a_{122}·a_{231}·b_{313}+a_{122}·a_{233}·a_{311}+a_{122}·b_{233}·a_{311}-a_{123}·a_{211}·a_{332}-b_{123}·a_{211}·a_{332}$$

$$+a_{123}·a_{212}·a_{331}+b_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}+b_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}-b_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}+a_{131}·a_{212}·b_{323}-a_{131}·a_{213}·a_{322}-a_{131}·b_{213}·a_{322}-a_{131}·a_{222}·a_{313}-a_{131}·a_{222}·b_{313}$$

$$+a_{131}·a_{223}·a_{312}+a_{131}·b_{223}·a_{312}-a_{132}·a_{211}·a_{323}-a_{132}·a_{211}·b_{323}+a_{132}·a_{213}·a_{321}+a_{132}·b_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}+a_{132}·a_{221}·b_{313}-a_{132}·a_{223}·a_{311}-a_{132}·b_{223}·a_{311}+a_{133}·a_{211}·a_{322}+b_{133}·a_{211}·a_{322}$$

$$-a_{133}·a_{212}·a_{321}-b_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}-b_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}+b_{133}·a_{222}·a_{311}$$

If we compare results of above equations, we can see that we have the same result in both cases.

8. For plan $k=2$: Let A and B be cubic-matrices of order 3, where all elements on the plan $k=1$ and $k=3$ are identical in both matrices, then we have:

$$det\left[A_{3\times 3\times 3}\right]+det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$+det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{b}_{112}& b_{122}& b_{132}\\ b_{212}& b_{222}& b_{232}\\ b_{312}& b_{322}& b_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=\{a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}\}$$

$$+\{a_{111}·b_{222}·a_{333}-a_{111}·b_{232}·a_{323}-a_{111}·a_{223}·b_{332}+a_{111}·a_{233}·b_{322}-b_{112}·a_{221}·a_{333}+b_{112}·a_{223}·a_{331}$$

$$+b_{112}·a_{231}·a_{323}-b_{112}·a_{233}·a_{321}+a_{113}·a_{221}·b_{332}-a_{113}·b_{222}·a_{331}-a_{113}·a_{231}·b_{322}+a_{113}·b_{232}·a_{321}$$

$$-a_{121}·b_{212}·a_{333}+a_{121}·a_{213}·b_{332}+a_{121}·b_{232}·a_{313}-a_{121}·a_{233}·b_{312}+b_{122}·a_{211}·a_{333}-b_{122}·a_{213}·a_{331}$$

$$-b_{122}·a_{231}·a_{313}+b_{122}·a_{233}·a_{311}-a_{123}·a_{211}·b_{332}+a_{123}·b_{212}·a_{331}+a_{123}·a_{231}·b_{312}-a_{123}·b_{232}·a_{311}$$

$$+a_{131}·b_{212}·a_{323}-a_{131}·a_{213}·b_{322}-a_{131}·b_{222}·a_{313}+a_{131}·a_{223}·b_{312}-b_{132}·a_{211}·a_{323}+b_{132}·a_{213}·a_{321}$$

$$+b_{132}·a_{221}·a_{313}-b_{132}·a_{223}·a_{311}+a_{133}·a_{211}·b_{322}-a_{133}·b_{212}·a_{321}-a_{133}·a_{221}·b_{312}+a_{133}·b_{222}·a_{311}\}.$$

Whereas,

$$det[A_{3\times 3\times 3}+B_{3\times 3\times 3}]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}+b_{112}& a_{122}+b_{122}& a_{132}+b_{132}\\ a_{212}+b_{212}& a_{222}+b_{222}& a_{232}+b_{232}\\ a_{312}+b_{312}& a_{322}+b_{322}& a_{332}+b_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=a_{111}·(a_{222}+b_{222})·a_{333}-a_{111}·(a_{232}+b_{232})·a_{323}-a_{111}·a_{223}·(a_{332}+b_{332})+a_{111}·a_{233}·(a_{322}+b_{322})$$

$$-(a_{112}+b_{112})·a_{221}·a_{333}+(a_{112}+b_{112})·a_{223}·a_{331}+(a_{112}+b_{112})·a_{231}·a_{323}-(a_{112}+b_{112})·a_{233}·a_{321}$$

$$+a_{113}·a_{221}·(a_{332}+b_{332})-a_{113}·(a_{222}+b_{222})·a_{331}-a_{113}·a_{231}·(a_{322}+b_{322})+a_{113}·(a_{232}+b_{232})·a_{321}$$

$$-a_{121}·(a_{212}+b_{212})·a_{333}+a_{121}·a_{213}·(a_{332}+b_{332})+a_{121}·(a_{232}+b_{232})·a_{313}-a_{121}·a_{233}·(a_{312}+b_{312})$$

$$+(a_{122}+b_{122})·a_{211}·a_{333}-(a_{122}+b_{122})·a_{213}·a_{331}-(a_{122}+b_{122})·a_{231}·a_{313}+(a_{122}+b_{122})·a_{233}·a_{311}$$

$$-a_{123}·a_{211}·(a_{332}+b_{332})+a_{123}·(a_{212}+b_{212})·a_{331}+a_{123}·a_{231}·(a_{312}+b_{312})-a_{123}·(a_{232}+b_{232})·a_{311}$$

$$+a_{131}·(a_{212}+b_{212})·a_{323}-a_{131}·a_{213}·(a_{322}+b_{322})-a_{131}·(a_{222}+b_{222})·a_{313}+a_{131}·a_{223}·(a_{312}+b_{312})$$

$$-(a_{132}+b_{132})·a_{211}·a_{323}+(a_{132}+b_{132})·a_{213}·a_{321}+(a_{132}+b_{132})·a_{221}·a_{313}-(a_{132}+b_{132})·a_{223}·a_{311}$$

$$+a_{133}·a_{211}·(a_{322}+b_{322})-a_{133}·(a_{212}+b_{212})·a_{321}-a_{133}·a_{221}·(a_{312}+b_{312})+a_{133}·(a_{222}+b_{222})·a_{311}$$

$$=a_{111}·a_{222}·a_{333}+a_{111}·b_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·b_{232}·a_{323}-a_{111}·a_{223}·a_{332}-a_{111}·a_{223}·b_{332}$$

$$+a_{111}·a_{233}·a_{322}+a_{111}·a_{233}·b_{322}-a_{112}·a_{221}·a_{333}-b_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}+b_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}+b_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}-b_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}+a_{113}·a_{221}·b_{332}$$

$$-a_{113}·a_{222}·a_{331}-a_{113}·b_{222}·a_{331}-a_{113}·a_{231}·a_{322}-a_{113}·a_{231}·b_{322}+a_{113}·a_{232}·a_{321}+a_{113}·b_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}-a_{121}·b_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{213}·b_{332}+a_{121}·a_{232}·a_{313}+a_{121}·b_{232}·a_{313}$$

$$-a_{121}·a_{233}·a_{312}-a_{121}·a_{233}·b_{312}+a_{122}·a_{211}·a_{333}+b_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}-b_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}-b_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}+b_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}-a_{123}·a_{211}·b_{332}$$

$$+a_{123}·a_{212}·a_{331}+a_{123}·b_{212}·a_{331}+a_{123}·a_{231}·a_{312}+a_{123}·a_{231}·b_{312}-a_{123}·a_{232}·a_{311}-a_{123}·b_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}+a_{131}·b_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{213}·b_{322}-a_{131}·a_{222}·a_{313}-a_{131}·b_{222}·a_{313}$$

$$+a_{131}·a_{223}·a_{312}+a_{131}·a_{223}·b_{312}-a_{132}·a_{211}·a_{323}-b_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}+b_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}+b_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}-b_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}+a_{133}·a_{211}·b_{322}$$

$$-a_{133}·a_{212}·a_{321}-a_{133}·b_{212}·a_{321}-a_{133}·a_{221}·a_{312}-a_{133}·a_{221}·b_{312}+a_{133}·a_{222}·a_{311}+a_{133}·b_{222}·a_{311}$$

If we compare results of above equations, we can see that we have the same result in both cases.

9. For plan $k=3$: Let A and B be cubic-matrices of order 3, where all elements on the plan $k=2$ and $k=3$ are identical in both matrices, then we have:

$$det\left[A_{3\times 3\times 3}\right]+det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$+det\left(\begin{array}{c}\left.\begin{array}{ccc}{b}_{111}& b_{121}& b_{131}\\ b_{211}& b_{221}& b_{231}\\ b_{311}& b_{321}& b_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=\{a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}\}$$

$$+\{b_{111}·a_{222}·a_{333}-b_{111}·a_{232}·a_{323}-b_{111}·a_{223}·a_{332}+b_{111}·a_{233}·a_{322}-a_{112}·b_{221}·a_{333}+a_{112}·a_{223}·b_{331}$$

$$+a_{112}·b_{231}·a_{323}-a_{112}·a_{233}·b_{321}+a_{113}·b_{221}·a_{332}-a_{113}·a_{222}·b_{331}-a_{113}·b_{231}·a_{322}+a_{113}·a_{232}·b_{321}$$

$$-b_{121}·a_{212}·a_{333}+b_{121}·a_{213}·a_{332}+b_{121}·a_{232}·a_{313}-b_{121}·a_{233}·a_{312}+a_{122}·b_{211}·a_{333}-a_{122}·a_{213}·b_{331}$$

$$-a_{122}·b_{231}·a_{313}+a_{122}·a_{233}·b_{311}-a_{123}·b_{211}·a_{332}+a_{123}·a_{212}·b_{331}+a_{123}·b_{231}·a_{312}-a_{123}·a_{232}·b_{311}$$

$$+b_{131}·a_{212}·a_{323}-b_{131}·a_{213}·a_{322}-b_{131}·a_{222}·a_{313}+b_{131}·a_{223}·a_{312}-a_{132}·b_{211}·a_{323}+a_{132}·a_{213}·b_{321}$$

$$+a_{132}·b_{221}·a_{313}-a_{132}·a_{223}·b_{311}+a_{133}·b_{211}·a_{322}-a_{133}·a_{212}·b_{321}-a_{133}·b_{221}·a_{312}+a_{133}·a_{222}·b_{311}\}.$$

Whereas,

$$det[A_{3\times 3\times 3}+B_{3\times 3\times 3}]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}+b_{111}& a_{121}+b_{121}& a_{131}+b_{131}\\ a_{211}+b_{211}& a_{221}+b_{221}& a_{231}+b_{231}\\ a_{311}+b_{311}& a_{321}+b_{321}& a_{331}+b_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=(a_{111}+b_{111})·a_{222}·a_{333}-(a_{111}+b_{111})·a_{232}·a_{323}-(a_{111}+b_{111})·a_{223}·a_{332}+(a_{111}+b_{111})·a_{233}·a_{322}$$

$$-a_{112}·(a_{221}+b_{221})·a_{333}+a_{112}·a_{223}·(a_{331}+b_{331})+a_{112}·(a_{231}+b_{231})·a_{323}-a_{112}·a_{233}·(a_{321}+b_{321})$$

$$+a_{113}·(a_{221}+b_{221})·a_{332}-a_{113}·a_{222}·(a_{331}+b_{331})-a_{113}·(a_{231}+b_{231})·a_{322}+a_{113}·a_{232}·(a_{321}+b_{321})$$

$$-(a_{121}+b_{121})·a_{212}·a_{333}+(a_{121}+b_{121})·a_{213}·a_{332}+(a_{121}+b_{121})·a_{232}·a_{313}-(a_{121}+b_{121})·a_{233}·a_{312}$$

$$+a_{122}·(a_{211}+b_{211})·a_{333}-a_{122}·a_{213}·(a_{331}+b_{331})-a_{122}·(a_{231}+b_{231})·a_{313}+a_{122}·a_{233}·(a_{311}+b_{311})$$

$$-a_{123}·(a_{211}+b_{211})·a_{332}+a_{123}·a_{212}·(a_{331}+b_{331})+a_{123}·(a_{231}+b_{231})·a_{312}-a_{123}·a_{232}·(a_{311}+b_{311})$$

$$+(a_{131}+b_{131})·a_{212}·a_{323}-(a_{131}+b_{131})·a_{213}·a_{322}-(a_{131}+b_{131})·a_{222}·a_{313}+(a_{131}+b_{131})·a_{223}·a_{312}$$

$$-a_{132}·(a_{211}+b_{211})·a_{323}+a_{132}·a_{213}·(a_{321}+b_{321})+a_{132}·(a_{221}+b_{221})·a_{313}-a_{132}·a_{223}·(a_{311}+b_{311})$$

$$+a_{133}·(a_{211}+b_{211})·a_{322}-a_{133}·a_{212}·(a_{321}+b_{321})-a_{133}·(a_{221}+b_{221})·a_{312}+a_{133}·a_{222}·(a_{311}+b_{311})$$

$$=a_{111}·a_{222}·a_{333}+b_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-b_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}-b_{111}·a_{223}·a_{332}$$

$$+a_{111}·a_{233}·a_{322}+b_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}-a_{112}·b_{221}·a_{333}+a_{112}·a_{223}·a_{331}+a_{112}·a_{223}·b_{331}$$

$$+a_{112}·a_{231}·a_{323}+a_{112}·b_{231}·a_{323}-a_{112}·a_{233}·a_{321}-a_{112}·a_{233}·b_{321}+a_{113}·a_{221}·a_{332}+a_{113}·b_{221}·a_{332}$$

$$-a_{113}·a_{222}·a_{331}-a_{113}·a_{222}·b_{331}-a_{113}·a_{231}·a_{322}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}+a_{113}·a_{232}·b_{321}$$

$$-a_{121}·a_{212}·a_{333}-b_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+b_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}+b_{121}·a_{232}·a_{313}$$

$$-a_{121}·a_{233}·a_{312}-b_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}+a_{122}·b_{211}·a_{333}-a_{122}·a_{213}·a_{331}-a_{122}·a_{213}·b_{331}$$

$$-a_{122}·a_{231}·a_{313}-a_{122}·b_{231}·a_{313}+a_{122}·a_{233}·a_{311}+a_{122}·a_{233}·b_{311}-a_{123}·a_{211}·a_{332}-a_{123}·b_{211}·a_{332}$$

$$+a_{123}·a_{212}·a_{331}+a_{123}·a_{212}·b_{331}+a_{123}·a_{231}·a_{312}+a_{123}·b_{231}·a_{312}-a_{123}·a_{232}·a_{311}-a_{123}·a_{232}·b_{311}$$

$$+a_{131}·a_{212}·a_{323}+b_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-b_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}-b_{131}·a_{222}·a_{313}$$

$$+a_{131}·a_{223}·a_{312}+b_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}-a_{132}·b_{211}·a_{323}+a_{132}·a_{213}·a_{321}+a_{132}·a_{213}·b_{321}$$

$$+a_{132}·a_{221}·a_{313}+a_{132}·b_{221}·a_{313}-a_{132}·a_{223}·a_{311}-a_{132}·a_{223}·b_{311}+a_{133}·a_{211}·a_{322}+a_{133}·b_{211}·a_{322}$$

$$-a_{133}·a_{212}·a_{321}-a_{133}·a_{212}·b_{321}-a_{133}·a_{221}·a_{312}-a_{133}·b_{221}·a_{312}+a_{133}·a_{222}·a_{311}+a_{133}·a_{222}·b_{311}.$$

If we compare results of above equations, we can see that we have the same result in both cases.

□

Theorem 2.

Let it be the cubic matrix B, which is formed by multiplying a plane (in the indices j and k, it does not work in the index i) of the matrix A by a scalar α, and addition another plane, then we have:

$$det\left(A\right)=det\left(B\right)$$

Proof. Case 1. The cubic-matrix A of order 2, (and B has order 2), we will proof the case 1 for each "vertical page" and "vertical layer", as following:

1. For plan $j=1$: Let us add first vertical page to second vertical page while multiplying by a scalar $\alpha$.

$$\begin{array}{cc}\hfill det\left[A_{2\times 2\times 2}\right]& =det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)\hfill \\ & =a_{111}·a_{222}-a_{112}·a_{221}-a_{121}·a_{212}+a_{122}·a_{211}.\hfill \end{array}$$

Whereas,

$$\begin{array}{cc}\hfill det\left[B_{2\times 2\times 2}\right]& =det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}+\alpha ·a_{111}\\ a_{211}& a_{221}+\alpha ·a_{211}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}+\alpha ·a_{112}\\ a_{212}& a_{222}+\alpha ·a_{212}\end{array}\end{array}\right)\hfill \\ & =a_{111}·(a_{222}+\alpha ·a_{212})-(a_{122}+\alpha ·a_{112})·a_{221}-(a_{121}+\alpha ·a_{111})·a_{212}+a_{122}·(a_{221}+\alpha ·a_{211}).\hfill \end{array}$$

After expanding further we get the following result

$$det\left[B_{2\times 2\times 2}\right]=a_{111}·a_{222}-a_{112}·a_{221}-a_{121}·a_{212}+a_{122}·a_{211}$$

If we compare results of above equations, we can see that we have the same result in both cases.

2. For plan $j=2$: Let us add second vertical page to first vertical page while multiplying by a scalar $\alpha$.

$$\begin{array}{cc}\hfill det\left[A_{2\times 2\times 2}\right]& =det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)\hfill \\ & =a_{111}·a_{222}-a_{112}·a_{221}-a_{121}·a_{212}+a_{122}·a_{211}.\hfill \end{array}$$

Whereas,

$$\begin{array}{cc}\hfill det\left[B_{2\times 2\times 2}\right]& =det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}+\alpha ·a_{121}& a_{121}\\ a_{211}+\alpha ·a_{221}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}+\alpha ·a_{122}& a_{122}\\ a_{212}+\alpha ·a_{222}& a_{222}\end{array}\end{array}\right)\hfill \\ & =(a_{111}+\alpha ·a_{121})·a_{222}-(a_{112}+\alpha ·a_{122})·a_{221}-a_{121}·(a_{212}+\alpha ·a_{222})+a_{122}·(a_{211}+\alpha ·a_{221}).\hfill \end{array}$$

After expanding further we get the following result

$$det\left[B_{2\times 2\times 2}\right]=a_{111}·a_{222}-a_{112}·a_{221}-a_{121}·a_{212}+a_{122}·a_{211}$$

If we compare results of above equations, we can see that we have the same result in both cases.

3. For plan $k=1$: Let us add first vertical page to second vertical layer while multiplying by a scalar $\alpha$.

$$\begin{array}{cc}\hfill det\left[A_{2\times 2\times 2}\right]& =det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)\hfill \\ & =a_{111}·a_{222}-a_{112}·a_{221}-a_{121}·a_{212}+a_{122}·a_{211}.\hfill \end{array}$$

Whereas,

$$\begin{array}{cc}\hfill det\left[B_{2\times 2\times 2}\right]& =det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}+\alpha ·a_{111}& a_{122}+\alpha ·a_{121}\\ a_{212}+\alpha ·a_{211}& a_{222}+\alpha ·a_{221}\end{array}\end{array}\right)\hfill \\ & =a_{111}·(a_{222}+\alpha ·a_{221})-(a_{112}+\alpha ·a_{111})·a_{221}-a_{121}·(a_{212}+\alpha ·a_{211})+(a_{122}+\alpha ·a_{121})·a_{211}.\hfill \end{array}$$

After expanding further we get the following result

$$det\left[B_{2\times 2\times 2}\right]=a_{111}·a_{222}-a_{112}·a_{221}-a_{121}·a_{212}+a_{122}·a_{211}$$

If we compare results of above equations, we can see that we have the same result in both cases.

4. For plan $k=2$: Let us add second vertical page to first vertical layer while multiplying by a scalar $\alpha$.

$$\begin{array}{cc}\hfill det\left[A_{2\times 2\times 2}\right]& =det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)\hfill \\ & =a_{111}·a_{222}-a_{112}·a_{221}-a_{121}·a_{212}+a_{122}·a_{211}.\hfill \end{array}$$

Whereas,

$$\begin{array}{cc}\hfill det\left[B_{2\times 2\times 2}\right]& =det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}+\alpha ·a_{112}& a_{121}+\alpha ·a_{122}\\ a_{211}+\alpha ·a_{212}& a_{221}+\alpha ·a_{222}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{122}\\ a_{212}& a_{222}\end{array}\end{array}\right)\hfill \\ & =(a_{111}+\alpha ·a_{112})·a_{222}-a_{112}·(a_{221}+\alpha ·a_{222})-(a_{121}+\alpha ·a_{122})·a_{212}+a_{122}·(a_{211}+\alpha ·a_{212}).\hfill \end{array}$$

After expanding further we get the following result

$$det\left[B_{2\times 2\times 2}\right]=a_{111}·a_{222}-a_{112}·a_{221}-a_{121}·a_{212}+a_{122}·a_{211}.$$

If we compare results of above equations, we can see that we have the same result in both cases.

Case 2. The cubic-matrix A of order 3, (and B has order 3), we will proof the case 2 for each "vertical page" and "vertical layer", as following:

1. For plan $j=1$: Let us add first vertical page to second vertical page while multiplying by a scalar $\alpha$.

$$det\left[A_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}.$$

Whereas,

$$det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}+\alpha ·a_{111}& a_{131}\\ a_{211}& a_{221}+\alpha ·a_{211}& a_{231}\\ a_{311}& a_{321}+\alpha ·a_{311}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}+\alpha ·a_{112}& a_{132}\\ a_{212}& a_{222}+\alpha ·a_{212}& a_{232}\\ a_{312}& a_{322}+\alpha ·a_{312}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}+\alpha ·a_{113}& a_{133}\\ a_{213}& a_{223}+\alpha ·a_{213}& a_{233}\\ a_{313}& a_{323}+\alpha ·a_{313}& a_{333}\end{array}\right.\end{array}\right)$$

$$=a_{111}·(a_{222}+\alpha ·a_{212})·a_{333}-a_{111}·a_{232}·(a_{323}+\alpha ·a_{313})-a_{111}·(a_{223}+\alpha ·a_{213})·a_{332}+a_{111}·a_{233}·(a_{322}+\alpha ·a_{312})$$

$$-a_{112}·(a_{221}+\alpha ·a_{211})·a_{333}+a_{112}·(a_{223}+\alpha ·a_{213})·a_{331}+a_{112}·a_{231}·(a_{323}+\alpha ·a_{313})-a_{112}·a_{233}·(a_{321}+\alpha ·a_{311})$$

$$+a_{113}·(a_{221}+\alpha ·a_{211})·a_{332}-a_{113}·(a_{222}+\alpha ·a_{212})·a_{331}-a_{113}·a_{231}·(a_{322}+\alpha ·a_{312})+a_{113}·a_{232}·(a_{321}+\alpha ·a_{311})$$

$$-(a_{121}+\alpha ·a_{111})·a_{212}·a_{333}+(a_{121}+\alpha ·a_{111})·a_{213}·a_{332}+(a_{121}+\alpha ·a_{111})·a_{232}·a_{313}-(a_{121}+\alpha ·a_{111})·a_{233}·a_{312}$$

$$+(a_{122}+\alpha ·a_{112})·a_{211}·a_{333}-(a_{122}+\alpha ·a_{112})·a_{213}·a_{331}-(a_{122}+\alpha ·a_{112})·a_{231}·a_{313}+(a_{122}+\alpha ·a_{112})·a_{233}·a_{311}$$

$$-(a_{123}+\alpha ·a_{113})·a_{211}·a_{332}+(a_{123}+\alpha ·a_{113})·a_{212}·a_{331}+(a_{123}+\alpha ·a_{113})·a_{231}·a_{312}-(a_{123}+\alpha ·a_{113})·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·(a_{323}+\alpha ·a_{313})-a_{131}·a_{213}·(a_{322}+\alpha ·a_{312})-a_{131}·(a_{222}+\alpha ·a_{212})·a_{313}+a_{131}·(a_{223}+\alpha ·a_{213})·a_{312}$$

$$-a_{132}·a_{211}·(a_{323}+\alpha ·a_{313})+a_{132}·a_{213}·(a_{321}+\alpha ·a_{311})+a_{132}·(a_{221}+\alpha ·a_{211})·a_{313}-a_{132}·(a_{223}+\alpha ·a_{213})·a_{311}$$

$$+a_{133}·a_{211}·(a_{322}+\alpha ·a_{312})-a_{133}·a_{212}·(a_{321}+\alpha ·a_{311})-a_{133}·(a_{221}+\alpha ·a_{211})·a_{312}+a_{133}·(a_{222}+\alpha ·a_{212})·a_{311}.$$

After expanding further we get the following result

$$=a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}.$$

If we compare results of above equations, we can see that we have the same result in both cases.

2. For plan $j=2$: Let us add first vertical page to third vertical page while multiplying by a scalar $\alpha$.

$$det\left[A_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}.$$

Whereas,

$$det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}+\alpha ·a_{111}\\ a_{211}& a_{221}& a_{231}+\alpha ·a_{211}\\ a_{311}& a_{321}& a_{331}+\alpha ·a_{311}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}+\alpha ·a_{112}\\ a_{212}& a_{222}& a_{232}+\alpha ·a_{212}\\ a_{312}& a_{322}& a_{332}+\alpha ·a_{312}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}+\alpha ·a_{113}\\ a_{213}& a_{223}& a_{233}+\alpha ·a_{213}\\ a_{313}& a_{323}& a_{333}+\alpha ·a_{313}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·(a_{333}+\alpha ·a_{313})-a_{111}·(a_{232}+\alpha ·a_{212})·a_{323}-a_{111}·a_{223}·(a_{332}+\alpha ·a_{312})+a_{111}·(a_{233}+\alpha ·a_{213})·a_{322}$$

$$-a_{112}·a_{221}·(a_{333}+\alpha ·a_{313})+a_{112}·a_{223}·(a_{331}+\alpha ·a_{311})+a_{112}·(a_{231}+\alpha ·a_{211})·a_{323}-a_{112}·(a_{233}+\alpha ·a_{213})·a_{321}$$

$$+a_{113}·a_{221}·(a_{332}+\alpha ·a_{312})-a_{113}·a_{222}·(a_{331}+\alpha ·a_{311})-a_{113}·(a_{231}+\alpha ·a_{211})·a_{322}+a_{113}·(a_{232}+\alpha ·a_{212})·a_{321}$$

$$-a_{121}·a_{212}·(a_{333}+\alpha ·a_{313})+a_{121}·a_{213}·(a_{332}+\alpha ·a_{312})+a_{121}·(a_{232}+\alpha ·a_{212})·a_{313}-a_{121}·(a_{233}+\alpha ·a_{213})·a_{312}$$

$$+a_{122}·a_{211}·(a_{333}+\alpha ·a_{313})-a_{122}·a_{213}·(a_{331}+\alpha ·a_{311})-a_{122}·(a_{231}+\alpha ·a_{211})·a_{313}+a_{122}·(a_{233}+\alpha ·a_{213})·a_{311}$$

$$-a_{123}·a_{211}·(a_{332}+\alpha ·a_{312})+a_{123}·a_{212}·(a_{331}+\alpha ·a_{311})+a_{123}·(a_{231}+\alpha ·a_{211})·a_{312}-a_{123}·(a_{232}+\alpha ·a_{212})·a_{311}$$

$$+(a_{131}+\alpha ·a_{111})·a_{212}·a_{323}-(a_{131}+\alpha ·a_{111})·a_{213}·a_{322}-(a_{131}+\alpha ·a_{111})·a_{222}·a_{313}+(a_{131}+\alpha ·a_{111})·a_{223}·a_{312}$$

$$-(a_{132}+\alpha ·a_{112})·a_{211}·a_{323}+(a_{132}+\alpha ·a_{112})·a_{213}·a_{321}+(a_{132}+\alpha ·a_{112})·a_{221}·a_{313}-(a_{132}+\alpha ·a_{112})·a_{223}·a_{311}$$

$$+(a_{133}+\alpha ·a_{113})·a_{211}·a_{322}-(a_{133}+\alpha ·a_{113})·a_{212}·a_{321}-(a_{133}+\alpha ·a_{113})·a_{221}·a_{312}+(a_{133}+\alpha ·a_{113})·a_{222}·a_{311}.$$

After expanding further we get the following result

$$=a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}.$$

If we compare results of above equations, we can see that we have the same result in both cases.

3. For plan $j=3$: Let us add second vertical page to third vertical page while multiplying by a scalar $\alpha$.

$$det\left[A_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}.$$

Whereas,

$$det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}+\alpha ·a_{121}\\ a_{211}& a_{221}& a_{231}+\alpha ·a_{221}\\ a_{311}& a_{321}& a_{331}+\alpha ·a_{321}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}+\alpha ·a_{122}\\ a_{212}& a_{222}& a_{232}+\alpha ·a_{222}\\ a_{312}& a_{322}& a_{332}+\alpha ·a_{322}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}+\alpha ·a_{123}\\ a_{213}& a_{223}& a_{233}+\alpha ·a_{223}\\ a_{313}& a_{323}& a_{333}+\alpha ·a_{323}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·(a_{333}+\alpha ·a_{323})-a_{111}·(a_{232}+\alpha ·a_{222})·a_{323}-a_{111}·a_{223}·(a_{332}+\alpha ·a_{312})+a_{111}·(a_{233}+\alpha ·a_{223})·a_{322}$$

$$-a_{112}·a_{221}·(a_{333}+\alpha ·a_{323})+a_{112}·a_{223}·(a_{331}+\alpha ·a_{321})+a_{112}·(a_{231}+\alpha ·a_{211})·a_{323}-a_{112}·(a_{233}+\alpha ·a_{223})·a_{321}$$

$$+a_{113}·a_{221}·(a_{332}+\alpha ·a_{322})-a_{113}·a_{222}·(a_{331}+\alpha ·a_{321})-a_{113}·(a_{231}+\alpha ·a_{211})·a_{322}+a_{113}·(a_{232}+\alpha ·a_{222})·a_{321}$$

$$-a_{121}·a_{212}·(a_{333}+\alpha ·a_{323})+a_{121}·a_{213}·(a_{332}+\alpha ·a_{322})+a_{121}·(a_{232}+\alpha ·a_{212})·a_{313}-a_{121}·(a_{233}+\alpha ·a_{223})·a_{312}$$

$$+a_{122}·a_{211}·(a_{333}+\alpha ·a_{323})-a_{122}·a_{213}·(a_{331}+\alpha ·a_{321})-a_{122}·(a_{231}+\alpha ·a_{211})·a_{313}+a_{122}·(a_{233}+\alpha ·a_{223})·a_{311}$$

$$-a_{123}·a_{211}·(a_{332}+\alpha ·a_{322})+a_{123}·a_{212}·(a_{331}+\alpha ·a_{321})+a_{123}·(a_{231}+\alpha ·a_{211})·a_{312}-a_{123}·(a_{232}+\alpha ·a_{222})·a_{311}$$

$$+(a_{131}+\alpha ·a_{121})·a_{212}·a_{323}-(a_{131}+\alpha ·a_{121})·a_{213}·a_{322}-(a_{131}+\alpha ·a_{121})·a_{222}·a_{313}+(a_{131}+\alpha ·a_{121})·a_{223}·a_{312}$$

$$-(a_{132}+\alpha ·a_{122})·a_{211}·a_{323}+(a_{132}+\alpha ·a_{122})·a_{213}·a_{321}+(a_{132}+\alpha ·a_{122})·a_{221}·a_{313}-(a_{132}+\alpha ·a_{122})·a_{223}·a_{311}$$

$$+(a_{133}+\alpha ·a_{123})·a_{211}·a_{322}-(a_{133}+\alpha ·a_{123})·a_{212}·a_{321}-(a_{133}+\alpha ·a_{123})·a_{221}·a_{312}+(a_{133}+\alpha ·a_{123})·a_{222}·a_{311}.$$

After expanding further we get the following result

$$=a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}.$$

If we compare results of above equations, we can see that we have the same result in both cases.

4. For plan $k=1$: Let us add first vertical layer to second vertical layer while multiplying by a scalar $\alpha$.

$$det\left[A_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}.$$

Whereas,

$$det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}+\alpha ·a_{111}& a_{122}+\alpha ·a_{121}& a_{132}+\alpha ·a_{131}\\ a_{212}+\alpha ·a_{211}& a_{222}+\alpha ·a_{221}& a_{232}+\alpha ·a_{231}\\ a_{312}+\alpha ·a_{311}& a_{322}+\alpha ·a_{321}& a_{332}+\alpha ·a_{331}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=a_{111}·(a_{222}+\alpha ·a_{221})·a_{333}-a_{111}·(a_{232}+\alpha ·a_{231})·a_{323}-a_{111}·a_{223}·(a_{332}+\alpha ·a_{331})+a_{111}·a_{233}·(a_{322}+\alpha ·a_{321})$$

$$-(a_{112}+\alpha ·a_{111})·a_{221}·a_{333}+(a_{112}+\alpha ·a_{111})·a_{223}·a_{331}+(a_{112}+\alpha ·a_{111})·a_{231}·a_{323}-(a_{112}+\alpha ·a_{111})·a_{233}·a_{321}$$

$$+a_{113}·a_{221}·(a_{332}+\alpha ·a_{331})-a_{113}·(a_{222}+\alpha ·a_{221})·a_{331}-a_{113}·a_{231}·(a_{322}+\alpha ·a_{321})+a_{113}·(a_{232}+\alpha ·a_{231})·a_{321}$$

$$-a_{121}·(a_{212}+\alpha ·a_{211})·a_{333}+a_{121}·a_{213}·(a_{332}+\alpha ·a_{331})+a_{121}·(a_{232}+\alpha ·a_{231})·a_{313}-a_{121}·a_{233}·(a_{312}+\alpha ·a_{311})$$

$$+(a_{122}+\alpha ·a_{121})·a_{211}·a_{333}-(a_{122}+\alpha ·a_{121})·a_{213}·a_{331}-(a_{122}+\alpha ·a_{121})·a_{231}·a_{313}+(a_{122}+\alpha ·a_{121})·a_{233}·a_{311}$$

$$-a_{123}·a_{211}·(a_{332}+\alpha ·a_{331})+a_{123}·(a_{212}+\alpha ·a_{211})·a_{331}+a_{123}·a_{231}·(a_{312}+\alpha ·a_{311})-a_{123}·(a_{232}+\alpha ·a_{231})·a_{311}$$

$$+a_{131}·(a_{212}+\alpha ·a_{211})·a_{323}-a_{131}·a_{213}·(a_{322}+\alpha ·a_{321})-a_{131}·(a_{222}+\alpha ·a_{221})·a_{313}+a_{131}·a_{223}·(a_{312}+\alpha ·a_{311})$$

$$-(a_{132}+\alpha ·a_{131})·a_{211}·a_{323}+(a_{132}+\alpha ·a_{131})·a_{213}·a_{321}+(a_{132}+\alpha ·a_{131})·a_{221}·a_{313}-(a_{132}+\alpha ·a_{131})·a_{223}·a_{311}$$

$$+a_{133}·a_{211}·(a_{322}+\alpha ·a_{321})-a_{133}·(a_{212}+\alpha ·a_{211})·a_{321}-a_{133}·a_{221}·(a_{312}+\alpha ·a_{311})+a_{133}·(a_{222}+\alpha ·a_{221})·a_{311}.$$

After expanding further we get the following result

$$=a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}.$$

If we compare results of above equations, we can see that we have the same result in both cases.

5. For plan $k=2$: Let us add first vertical layer to third vertical layer while multiplying by a scalar $\alpha$.

$$det\left[A_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}.$$

Whereas,

$$det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}+\alpha ·a_{111}& a_{123}+\alpha ·a_{121}& a_{133}+\alpha ·a_{131}\\ a_{213}+\alpha ·a_{211}& a_{223}+\alpha ·a_{221}& a_{233}+\alpha ·a_{231}\\ a_{313}+\alpha ·a_{311}& a_{323}+\alpha ·a_{321}& a_{333}+\alpha ·a_{331}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·(a_{333}+\alpha ·a_{331})-a_{111}·a_{232}·(a_{323}+\alpha ·a_{321})-a_{111}·(a_{223}+\alpha ·a_{221})·a_{332}+a_{111}·(a_{233}+\alpha ·a_{231})·a_{322}$$

$$-a_{112}·a_{221}·(a_{333}+\alpha ·a_{331})+a_{112}·(a_{223}+\alpha ·a_{221})·a_{331}+a_{112}·a_{231}·(a_{323}+\alpha ·a_{321})-a_{112}·(a_{233}+\alpha ·a_{231})·a_{321}$$

$$+(a_{113}+\alpha ·a_{111})·a_{221}·a_{332}-(a_{113}+\alpha ·a_{111})·a_{222}·a_{331}-(a_{113}+\alpha ·a_{111})·a_{231}·a_{322}+(a_{113}+\alpha ·a_{111})·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·(a_{333}+\alpha ·a_{331})+a_{121}·(a_{213}+\alpha ·a_{211})·a_{332}+a_{121}·a_{232}·(a_{313}+\alpha ·a_{311})-a_{121}·(a_{233}+\alpha ·a_{231})·a_{312}$$

$$+a_{122}·a_{211}·(a_{333}+\alpha ·a_{331})-a_{122}·(a_{213}+\alpha ·a_{211})·a_{331}-a_{122}·a_{231}·(a_{313}+\alpha ·a_{311})+a_{122}·(a_{233}+\alpha ·a_{231})·a_{311}$$

$$-(a_{123}+\alpha ·a_{121})·a_{211}·a_{332}+(a_{123}+\alpha ·a_{121})·a_{212}·a_{331}+(a_{123}+\alpha ·a_{121})·a_{231}·a_{312}-(a_{123}+\alpha ·a_{121})·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·(a_{323}+\alpha ·a_{321})-a_{131}·(a_{213}+\alpha ·a_{211})·a_{322}-a_{131}·a_{222}·(a_{313}+\alpha ·a_{311})+a_{131}·(a_{223}+\alpha ·a_{221})·a_{312}$$

$$-a_{132}·a_{211}·(a_{323}+\alpha ·a_{321})+a_{132}·(a_{213}+\alpha ·a_{211})·a_{321}+a_{132}·a_{221}·(a_{313}+\alpha ·a_{311})-a_{132}·(a_{223}+\alpha ·a_{221})·a_{311}$$

$$+(a_{133}+\alpha ·a_{131})·a_{211}·a_{322}-(a_{133}+\alpha ·a_{131})·a_{212}·a_{321}-(a_{133}+\alpha ·a_{131})·a_{221}·a_{312}+(a_{133}+\alpha ·a_{131})·a_{222}·a_{311}$$

$$+a_{133}·a_{211}·(a_{322}+\alpha ·a_{221})-a_{133}·(a_{212}+\alpha ·a_{211})·a_{321}-a_{133}·a_{221}·(a_{312}+\alpha ·a_{211})+a_{133}·(a_{222}+\alpha ·a_{221})·a_{311}.$$

After expanding further we get the following result,

$$=a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}.$$

If we compare results of above equations, we can see that we have the same result in both cases.

6. For plan $k=3$: Let us add first vertical layer to third vertical layer while multiplying by a scalar $\alpha$.

$$det\left[A_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}.$$

Whereas,

$$det\left[B_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}+\alpha ·a_{112}& a_{123}+\alpha ·a_{122}& a_{133}+\alpha ·a_{132}\\ a_{213}+\alpha ·a_{212}& a_{223}+\alpha ·a_{222}& a_{233}+\alpha ·a_{232}\\ a_{313}+\alpha ·a_{312}& a_{323}+\alpha ·a_{322}& a_{333}+\alpha ·a_{332}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·(a_{333}+\alpha ·a_{332})-a_{111}·a_{232}·(a_{323}+\alpha ·a_{322})-a_{111}·(a_{223}+\alpha ·a_{222})·a_{332}+a_{111}·(a_{233}+\alpha ·a_{232})·a_{322}$$

$$-a_{112}·a_{221}·(a_{333}+\alpha ·a_{332})+a_{112}·(a_{223}+\alpha ·a_{222})·a_{331}+a_{112}·a_{231}·(a_{323}+\alpha ·a_{322})-a_{112}·(a_{233}+\alpha ·a_{232})·a_{321}$$

$$+(a_{113}+\alpha ·a_{112})·a_{221}·a_{332}-(a_{113}+\alpha ·a_{112})·a_{222}·a_{331}-(a_{113}+\alpha ·a_{112})·a_{231}·a_{322}+(a_{113}+\alpha ·a_{112})·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·(a_{333}+\alpha ·a_{332})+a_{121}·(a_{213}+\alpha ·a_{212})·a_{332}+a_{121}·a_{232}·(a_{313}+\alpha ·a_{312})-a_{121}·(a_{233}+\alpha ·a_{232})·a_{312}$$

$$+a_{122}·a_{211}·(a_{333}+\alpha ·a_{332})-a_{122}·(a_{213}+\alpha ·a_{212})·a_{331}-a_{122}·a_{231}·(a_{313}+\alpha ·a_{312})+a_{122}·(a_{233}+\alpha ·a_{232})·a_{311}$$

$$-(a_{123}+\alpha ·a_{122})·a_{211}·a_{332}+(a_{123}+\alpha ·a_{122})·a_{212}·a_{331}+(a_{123}+\alpha ·a_{122})·a_{231}·a_{312}-(a_{123}+\alpha ·a_{122})·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·(a_{323}+\alpha ·a_{322})-a_{131}·(a_{213}+\alpha ·a_{212})·a_{322}-a_{131}·a_{222}·(a_{313}+\alpha ·a_{312})+a_{131}·(a_{223}+\alpha ·a_{222})·a_{312}$$

$$-a_{132}·a_{211}·(a_{323}+\alpha ·a_{322})+a_{132}·(a_{213}+\alpha ·a_{212})·a_{321}+a_{132}·a_{221}·(a_{313}+\alpha ·a_{312})-a_{132}·(a_{223}+\alpha ·a_{222})·a_{311}$$

$$+(a_{133}+\alpha ·a_{132})·a_{211}·a_{322}-(a_{133}+\alpha ·a_{132})·a_{212}·a_{321}-(a_{133}+\alpha ·a_{132})·a_{221}·a_{312}+(a_{133}+\alpha ·a_{132})·a_{222}·a_{311}$$

$$+a_{133}·a_{211}·(a_{322}+\alpha ·a_{222})-a_{133}·(a_{212}+\alpha ·a_{212})·a_{321}-a_{133}·a_{221}·(a_{312}+\alpha ·a_{212})+a_{133}·(a_{222}+\alpha ·a_{222})·a_{311}.$$

After expanding further we get the following result,

$$=a_{111}·a_{222}·a_{333}-a_{111}·a_{232}·a_{323}-a_{111}·a_{223}·a_{332}+a_{111}·a_{233}·a_{322}-a_{112}·a_{221}·a_{333}+a_{112}·a_{223}·a_{331}$$

$$+a_{112}·a_{231}·a_{323}-a_{112}·a_{233}·a_{321}+a_{113}·a_{221}·a_{332}-a_{113}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{113}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{333}+a_{121}·a_{213}·a_{332}+a_{121}·a_{232}·a_{313}-a_{121}·a_{233}·a_{312}+a_{122}·a_{211}·a_{333}-a_{122}·a_{213}·a_{331}$$

$$-a_{122}·a_{231}·a_{313}+a_{122}·a_{233}·a_{311}-a_{123}·a_{211}·a_{332}+a_{123}·a_{212}·a_{331}+a_{123}·a_{231}·a_{312}-a_{123}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{323}-a_{131}·a_{213}·a_{322}-a_{131}·a_{222}·a_{313}+a_{131}·a_{223}·a_{312}-a_{132}·a_{211}·a_{323}+a_{132}·a_{213}·a_{321}$$

$$+a_{132}·a_{221}·a_{313}-a_{132}·a_{223}·a_{311}+a_{133}·a_{211}·a_{322}-a_{133}·a_{212}·a_{321}-a_{133}·a_{221}·a_{312}+a_{133}·a_{222}·a_{311}.$$

If we compare results of above equations, we can see that we have the same result in both cases.

Remark 1.

This theorem does not hold for plan "Horizontal Layers".

Theorem 3.

Suppose that A is 3D Determinant with two identical "Vertical Pages" or two identical "Vertical Layers".

Then $\left|A\right|=0$.

Proof. Case 1. The cubic-matrix A of order 2 with two identical "Vertical Pages" or two identical "Vertical Layers", we will proof the case 1, as following:

1. For two identical "Vertical Pages":

$$\begin{array}{cc}\hfill det\left[A_{2\times 2\times 2}\right]& =det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{111}\\ a_{211}& a_{211}\end{array}\right|\begin{array}{cc}{a}_{112}& a_{112}\\ a_{212}& a_{212}\end{array}\end{array}\right)\hfill \\ & =a_{111}·a_{212}-a_{112}·a_{211}-a_{111}·a_{212}+a_{112}·a_{211}\hfill \\ & =0.\hfill \end{array}$$

2. For two identical "Vertical Layers":

$$\begin{array}{cc}\hfill det\left[A_{2\times 2\times 2}\right]& =det\left(\begin{array}{c}\left.\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\right|\begin{array}{cc}{a}_{111}& a_{121}\\ a_{211}& a_{221}\end{array}\end{array}\right)\hfill \\ & =a_{111}·a_{221}-a_{111}·a_{221}-a_{121}·a_{211}+a_{121}·a_{211}\hfill \\ & =0.\hfill \end{array}$$

Case 2. The cubic-matrix A of order 3 with two identical "Vertical Pages" or two identical "Vertical Layers", we will proof the case 1, as following:

1. For two identical "Vertical Pages", first "Vertical Page" identical to second "Vertical Page":

$$det\left[A_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{111}& a_{131}\\ a_{211}& a_{211}& a_{231}\\ a_{311}& a_{311}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{112}& a_{132}\\ a_{212}& a_{212}& a_{232}\\ a_{312}& a_{312}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{113}& a_{133}\\ a_{213}& a_{213}& a_{233}\\ a_{313}& a_{313}& a_{333}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{212}·a_{333}-a_{111}·a_{232}·a_{313}-a_{111}·a_{213}·a_{332}+a_{111}·a_{233}·a_{312}-a_{112}·a_{211}·a_{333}+a_{112}·a_{213}·a_{331}$$

$$+a_{112}·a_{231}·a_{313}-a_{112}·a_{233}·a_{311}+a_{113}·a_{211}·a_{332}-a_{113}·a_{212}·a_{331}-a_{113}·a_{231}·a_{312}+a_{113}·a_{232}·a_{311}$$

$$-a_{111}·a_{212}·a_{333}+a_{111}·a_{213}·a_{332}+a_{111}·a_{232}·a_{313}-a_{111}·a_{233}·a_{312}+a_{112}·a_{211}·a_{333}-a_{112}·a_{213}·a_{331}$$

$$-a_{112}·a_{231}·a_{313}+a_{112}·a_{233}·a_{311}-a_{113}·a_{211}·a_{332}+a_{113}·a_{212}·a_{331}+a_{113}·a_{231}·a_{312}-a_{113}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{313}-a_{131}·a_{213}·a_{312}-a_{131}·a_{212}·a_{313}+a_{131}·a_{213}·a_{312}-a_{132}·a_{211}·a_{313}+a_{132}·a_{213}·a_{311}$$

$$+a_{132}·a_{211}·a_{313}-a_{132}·a_{213}·a_{311}+a_{133}·a_{211}·a_{312}-a_{133}·a_{212}·a_{311}-a_{133}·a_{211}·a_{312}+a_{133}·a_{212}·a_{311}=0.$$

2. For two identical "Vertical Pages", first "Vertical Page" identical to third "Vertical Page":

$$det\left[A_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{111}\\ a_{211}& a_{221}& a_{211}\\ a_{311}& a_{321}& a_{311}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{112}\\ a_{212}& a_{222}& a_{212}\\ a_{312}& a_{322}& a_{312}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{113}\\ a_{213}& a_{223}& a_{213}\\ a_{313}& a_{323}& a_{313}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·a_{313}-a_{111}·a_{212}·a_{323}-a_{111}·a_{223}·a_{312}+a_{111}·a_{213}·a_{322}-a_{112}·a_{221}·a_{313}+a_{112}·a_{223}·a_{311}$$

$$+a_{112}·a_{211}·a_{323}-a_{112}·a_{213}·a_{321}+a_{113}·a_{221}·a_{312}-a_{113}·a_{222}·a_{311}-a_{113}·a_{211}·a_{322}+a_{113}·a_{212}·a_{321}$$

$$-a_{121}·a_{212}·a_{313}+a_{121}·a_{213}·a_{312}+a_{121}·a_{212}·a_{313}-a_{121}·a_{213}·a_{312}+a_{122}·a_{211}·a_{313}-a_{122}·a_{213}·a_{311}$$

$$-a_{122}·a_{211}·a_{313}+a_{122}·a_{213}·a_{311}-a_{123}·a_{211}·a_{312}+a_{123}·a_{212}·a_{311}+a_{123}·a_{211}·a_{312}-a_{123}·a_{212}·a_{311}$$

$$+a_{111}·a_{212}·a_{323}-a_{111}·a_{213}·a_{322}-a_{111}·a_{222}·a_{313}+a_{111}·a_{223}·a_{312}-a_{112}·a_{211}·a_{323}+a_{112}·a_{213}·a_{321}$$

$$+a_{112}·a_{221}·a_{313}-a_{112}·a_{223}·a_{311}+a_{113}·a_{211}·a_{322}-a_{113}·a_{212}·a_{321}-a_{113}·a_{221}·a_{312}+a_{113}·a_{222}·a_{311}=0.$$

3. For two identical "Vertical Pages", second "Vertical Page" identical to third "Vertical Page":

$$det\left[A_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{121}\\ a_{211}& a_{221}& a_{221}\\ a_{311}& a_{321}& a_{321}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{122}\\ a_{212}& a_{222}& a_{222}\\ a_{312}& a_{322}& a_{322}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{123}\\ a_{213}& a_{223}& a_{223}\\ a_{313}& a_{323}& a_{323}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·a_{323}-a_{111}·a_{222}·a_{323}-a_{111}·a_{223}·a_{322}+a_{111}·a_{223}·a_{322}-a_{112}·a_{221}·a_{323}+a_{112}·a_{223}·a_{321}$$

$$+a_{112}·a_{221}·a_{323}-a_{112}·a_{223}·a_{321}+a_{113}·a_{221}·a_{322}-a_{113}·a_{222}·a_{321}-a_{113}·a_{221}·a_{322}+a_{113}·a_{222}·a_{321}$$

$$-a_{121}·a_{212}·a_{323}+a_{121}·a_{213}·a_{322}+a_{121}·a_{222}·a_{313}-a_{121}·a_{223}·a_{312}+a_{122}·a_{211}·a_{323}-a_{122}·a_{213}·a_{321}$$

$$-a_{122}·a_{221}·a_{313}+a_{122}·a_{223}·a_{311}-a_{123}·a_{211}·a_{322}+a_{123}·a_{212}·a_{321}+a_{123}·a_{221}·a_{312}-a_{123}·a_{222}·a_{311}$$

$$+a_{121}·a_{212}·a_{323}-a_{121}·a_{213}·a_{322}-a_{121}·a_{222}·a_{313}+a_{121}·a_{223}·a_{312}-a_{122}·a_{211}·a_{323}+a_{122}·a_{213}·a_{321}$$

$$+a_{122}·a_{221}·a_{313}-a_{122}·a_{223}·a_{311}+a_{123}·a_{211}·a_{322}-a_{123}·a_{212}·a_{321}-a_{123}·a_{221}·a_{312}+a_{123}·a_{222}·a_{311}=0.$$

4. For two identical "Vertical Layers", first "Vertical Layer" identical to second "Vertical Layer":

$$det\left[A_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\left|\begin{array}{ccc}{a}_{113}& a_{123}& a_{133}\\ a_{213}& a_{223}& a_{233}\\ a_{313}& a_{323}& a_{333}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{221}·a_{333}-a_{111}·a_{231}·a_{323}-a_{111}·a_{223}·a_{331}+a_{111}·a_{233}·a_{321}-a_{111}·a_{221}·a_{333}+a_{111}·a_{223}·a_{331}$$

$$+a_{111}·a_{231}·a_{323}-a_{111}·a_{233}·a_{321}+a_{113}·a_{221}·a_{331}-a_{113}·a_{221}·a_{331}-a_{113}·a_{231}·a_{321}+a_{113}·a_{231}·a_{321}$$

$$-a_{121}·a_{211}·a_{333}+a_{121}·a_{213}·a_{331}+a_{121}·a_{231}·a_{313}-a_{121}·a_{233}·a_{311}+a_{121}·a_{211}·a_{333}-a_{121}·a_{213}·a_{331}$$

$$-a_{121}·a_{231}·a_{313}+a_{121}·a_{233}·a_{311}-a_{123}·a_{211}·a_{331}+a_{123}·a_{211}·a_{331}+a_{123}·a_{231}·a_{311}-a_{123}·a_{231}·a_{311}$$

$$+a_{131}·a_{211}·a_{323}-a_{131}·a_{213}·a_{321}-a_{131}·a_{221}·a_{313}+a_{131}·a_{223}·a_{311}-a_{131}·a_{211}·a_{323}+a_{131}·a_{213}·a_{321}$$

$$+a_{131}·a_{221}·a_{313}-a_{131}·a_{223}·a_{311}+a_{133}·a_{211}·a_{321}-a_{133}·a_{211}·a_{321}-a_{133}·a_{221}·a_{311}+a_{133}·a_{221}·a_{311}=0.$$

5. For two identical "Vertical Layers", first "Vertical Layer" identical to third "Vertical Layer":

$$det\left[A_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·a_{331}-a_{111}·a_{232}·a_{321}-a_{111}·a_{221}·a_{332}+a_{111}·a_{231}·a_{322}-a_{112}·a_{221}·a_{331}+a_{112}·a_{221}·a_{331}$$

$$+a_{112}·a_{231}·a_{321}-a_{112}·a_{231}·a_{321}+a_{111}·a_{221}·a_{332}-a_{111}·a_{222}·a_{331}-a_{111}·a_{231}·a_{322}+a_{111}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{331}+a_{121}·a_{211}·a_{332}+a_{121}·a_{232}·a_{311}-a_{121}·a_{231}·a_{312}+a_{122}·a_{211}·a_{331}-a_{122}·a_{211}·a_{331}$$

$$-a_{122}·a_{231}·a_{311}+a_{122}·a_{231}·a_{311}-a_{121}·a_{211}·a_{332}+a_{121}·a_{212}·a_{331}+a_{121}·a_{231}·a_{312}-a_{121}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{321}-a_{131}·a_{211}·a_{322}-a_{131}·a_{222}·a_{311}+a_{131}·a_{221}·a_{312}-a_{132}·a_{211}·a_{321}+a_{132}·a_{211}·a_{321}$$

$$+a_{132}·a_{221}·a_{311}-a_{132}·a_{221}·a_{311}+a_{131}·a_{211}·a_{322}-a_{131}·a_{212}·a_{321}-a_{131}·a_{221}·a_{312}+a_{131}·a_{222}·a_{311}=0.$$

6. For two identical "Vertical Layers", second "Vertical Layer" identical to third "Vertical Layer":

$$det\left[A_{3\times 3\times 3}\right]=det\left(\begin{array}{c}\left.\begin{array}{ccc}{a}_{111}& a_{121}& a_{131}\\ a_{211}& a_{221}& a_{231}\\ a_{311}& a_{321}& a_{331}\end{array}\right|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\left|\begin{array}{ccc}{a}_{112}& a_{122}& a_{132}\\ a_{212}& a_{222}& a_{232}\\ a_{312}& a_{322}& a_{332}\end{array}\right.\end{array}\right)$$

$$=a_{111}·a_{222}·a_{332}-a_{111}·a_{232}·a_{322}-a_{111}·a_{222}·a_{332}+a_{111}·a_{232}·a_{322}-a_{112}·a_{221}·a_{332}+a_{112}·a_{222}·a_{331}$$

$$+a_{112}·a_{231}·a_{322}-a_{112}·a_{232}·a_{321}+a_{112}·a_{221}·a_{332}-a_{112}·a_{222}·a_{331}-a_{113}·a_{231}·a_{322}+a_{112}·a_{232}·a_{321}$$

$$-a_{121}·a_{212}·a_{332}+a_{121}·a_{212}·a_{332}+a_{121}·a_{232}·a_{312}-a_{121}·a_{232}·a_{312}+a_{122}·a_{211}·a_{332}-a_{122}·a_{212}·a_{331}$$

$$-a_{122}·a_{231}·a_{312}+a_{122}·a_{232}·a_{311}-a_{122}·a_{211}·a_{332}+a_{122}·a_{212}·a_{331}+a_{122}·a_{231}·a_{312}-a_{122}·a_{232}·a_{311}$$

$$+a_{131}·a_{212}·a_{322}-a_{131}·a_{212}·a_{322}-a_{131}·a_{222}·a_{312}+a_{131}·a_{222}·a_{312}-a_{132}·a_{211}·a_{322}+a_{132}·a_{212}·a_{321}$$

$$+a_{132}·a_{221}·a_{312}-a_{132}·a_{222}·a_{311}+a_{132}·a_{211}·a_{322}-a_{132}·a_{212}·a_{321}-a_{132}·a_{221}·a_{312}+a_{132}·a_{222}·a_{311}=0.$$

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