Cramer's rule is a venerable but, except in special circumstances, mostly useless method for solving systems of equations. One excuse for teaching it is the following `gem' proof, for which we need some simple notation: let denote the matrix A with its column replaced by the column vector a. Then if Ax = b and ,
By the product rule for the determinant function, , we have that . A simple expansion along the row shows that , the entry in the unknown vector. This means that if , we have
This is Cramer's rule.
We note that unless A is sparse or has very special structure, this is an extremely costly way to solve Ax = b. This is `well-known', but is also a common error for people who write their first programs for solving linear systems to overlook.