Cross-Multiplication Method in Computing Determinants of n ×n Matrices

The determinant is one of the interesting and essential topics in matrix algebra for its applications and representation utility. In this paper, a novel, straightforward, and symmetry-based technique for manually computing the determinant of any n X n matrix is developed. The method is derived from Dodgson’s condensation method which involves the computation of determinants of four adjacent entries within the interior submatrix of a given matrix. By strategically applying elementary row (column) operations and the definition and properties of determinants, the new method proceeds without assigning an interior submatrix by fixing the nonzero first entries of two adjacent rows as pivot elements in computing the determinants of succeeding four entries. The process suggests a symmetric butterfly movement coined cross-multiplication, yields a more streamlined algorithm that is generalized through formulas, and employs a smaller number of operations and succeeding matrices than the existing methods.