Today I Learned

Some of the things I've learned every day since Oct 10, 2016

21: Computation of Determinants

Because the computation of matrix determinants by cofactor expansion (Laplace expansion) is computationally very expensive — for example, computing the determinant of an $n \times n$ matrix by this method requires over $n!$ multiplications — other methods are usually used.

For instance, the simple method of using Gaussian elimination to reduce the matrix to its upper-triangular form, then taking the product of the diagonal to obtain the determinant, is just $O(n^3)$ in multiplications. Other, even less expensive algorithms exist as well, such as the Bareiss algorithm.