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

Wed Jan 31 11:33:59 EST 1996