5.4 Cost of Solving a System of Linear Equations Using the Inverse

We saw in Section 2.4.6 that using Gaussian Elimination or Gauss-Jordan Elimination to find the inverse of an n x n matrix can be thought of as solving n linear systems having the same coefficient matrix, but different right-hand sides.

This means that the operation counts can be determined from the results in Section 5.3 by setting k (the number of right-hand sides) to n. So computing the inverse of a matrix using Gaussian Elimination requires multiplications and additions, while computing the inverse using Gauss-Jordan Elimination requires multiplications and additions.

Once we have computed the inverse, we need to perform a matrix-vector multiplication in order to determine the solution of the original problem. This adds another n² multiplications and n²-n additions.

Altogether, solving a system of n equations in n unknowns using an inverse computed using Gaussian Elimination requires multiplications and additions. And solving the same system with an inverse computed using Gauss-Jordan Elimination requires multiplications and additions.