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.

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