How Do We Use Cramers Rule Today?
The linear systems in science and engineering are typically way too big to be solved by Cramer’s Rule, e.g. 100,000 to 1,000,000 and looking toward 100,000,000. When all one needs to know is if a matrix A is singular or not, it is typical to replace the original matrix by a triangular matrix, its called a factorization, and to estimate the determinant of the original matrix by the product of the diagonal entries of the triangular matrix, also known as a factor matrix. Cramer Rule though can be used when working with formulas involving traces (sum of diagonal entries of a A) and derivatives, for instance when reconstructing A from several known b’s and x’s.
