5.5 Cost of Solving a System of Linear Equations Using Cramer's Rule

Cramer's Rule requires computation of (n+1) determinants in order to solve an n x n system of linear equations. For each determinant, n! multiplications and n! additions are required.^† Hence the cost of computing all the determinants is (n+1)(n!) = (n + 1)! multiplications and (n+1)(n!) = (n + 1)! additions.

We could reduce this number by computing the determinants by using Gaussian Elimination. The number of operations required to compute the determinant of an n x n matrix takes on the order of n³ operations. Since we have to determine n determinants, using Cramer's Rule with the determinants determined by Gaussian Elimination would require on the order of n⁴ operations. This is a lot more work than is required to solve the system using Gaussian Elimination.

^† See Exercise 15 at the end of this chapter.