Matrix Reduction by Cross-Multiplication as Alternative Method of Matrix Inversion and Solving Systems of Linear Equations

This paper presents a theoretical exploration of a computational technique on the strategic use of elementary row and column operations in matrix reduction to solve matrix inverse and systems of linear equations. Matrix reduction by cross-multiplication method is carried out with two succeeding rows of a matrix to form submatrices where a submatrix is one row and one column less than the preceding submatrix. When applied to matrix inversion and solving system of linear equations, the method proceeds in four main steps: matrix reduction to form upper triangular coefficient matrix, row and column flips to form lower triangular coefficient matrix, matrix reduction to form diagonal coefficient matrix, and specifying matrix inverse or solution set of the linear system. At the theoretical level, cross-multiplication method yields the least number of arithmetic operations and systematically reduces the matrices generated in matrix reduction compared to the existing methods. With the determinant formula derived using cross-multiplication, this alternative method connects the Gauss-Jordan reduction and determinant methods of matrix inversion, a feature not explicitly explained in current literature.