Determinant of the inverse of a matrix

linear-algebra, matrix

Solution

If you already have the Cholesky decomposition (A = L * L_t), then you just have

det(A) = det(L) * det(L_t) = sqr(det(L))

The L matrix is lower triangular so its determinant is the product of diagonal elements.

The Cholesky decomposition takes O(n^3) operations and the product of diagonal elements of L is only O(n). Gaussian elimination method (convert A to triangular matrix) would take O(n^3) and would suffer from possible numerical problems.

Finally, det(inv(A)) = 1/det(A).

Problem

How can I calculate the determinant of the inverse of a matrix using Cholesky decomposition. I have found this that it is not a good idea to directly calculate the determinant of matrix. So can anyone provide some insights?

Original source