Operations Counts

To solve a reasonably sized system of n equations in n unknowns, the amount of work the computer must do varies with algorithm. The operations the computer must do are mostly additions and multiplications, and the number that must be done is roughly:

• Cramer's rule: n! operations (prohibitive);
• Gaussian Elimination/LU-decomposition: reduce to echelon form at operations;
• Gauss-Jordan: reduce to row canonical form at operations;
• Matrix inversion: n³ operations.

Note that Gausian elimination can be much more efficient still for sparse matrices (i.e. matrices with a lot of zeros.) Always use the most specific algorithm for your matrix.