Modified Cramer’s Rule and its Application to Solve Linear Systems in WZ Factorization

Abstract

The proposed modified methods of Cramer's rule consider the column vector as well as the coefficient matrix concurrently in the linear system. The modified methods can be applied since Cramer's rule is typically known for solving the linear systems in $WZ$ factorization to yield Z-matrix. Then, we presented our results to show that there is no tangible difference in performance time between Cramer's rule and the modified methods in the factorization from improved versions of MATLAB. Additionally, the Frobenius norm of the modified methods in the factorization is better than using Cramer's rule irrespective of the version of MATLAB used.

Optimized Cramer’s Rule in WZ Factorization and Applications

“…Cramer's rule solves the 2 × 2 linear systems of W Z factorization under the nonsingularity constraint presumed for their determinants [8]. Though Cramer's rule is assumed to be less practical due to its setbacks, many modifications have been made on Cramer's rule to solve simple and large linear systems, see [5,29,41]. Due to round off errors which may become significant on problems with non-integer coefficients, Moler [35] then demonstrated that Cramer's rule is inadequate even for 2 × 2 linear systems.…”