In this paper, we introduce as a pedagogical strategy an internal division-free, straightforward, and symmetrically progressing algorithm in manually computing matrix inverse and solving systems of linear equations by revisiting the application of elementary row operations in the Gauss-Jordan reduction method and connecting it to the determinant method. The proposed cross-multiplication-flip (CMF) algorithm employs cross-multiplication similar to the butterfly movement in computing determinants as a strategic application of elementary row operations to efficiently reduce the rows and then applies flipping of rows and entries to put an upper triangular matrix into lower triangular form to continue the reduction process.

We formulate a more straightforward, symmetry-based techniques in manually computing the determinant of any n×n matrix by revisiting Dodgson’s condensation method and strategically applying elementary row operations and the definition and properties of determinants. The result yields a more simplified mnemonical algorithm than the usual reduction by elementary row operations and co-factor expansion and with minimal terminal divisions performed compared to Dodgson’s internal divisions.