## Computing determinants from matrix decompositions

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There are a few properties of a matrix that can make it easy to compute determinants.

1. The determinant of a triangular matrix is the product of the elements on the diagonal.
2. The determinant of a permutation matrix is (-1)**n where n is the number of permutations. Recall a permutation matrix is a matrix with a one in each row, and column, and zeros everywhere else.
3. The determinant of a product of matrices is equal to the product of the determinant of the matrices.

The LU decomposition computes three matrices such that $$A = P L U$$. Thus, $$\det A = \det P \det L \det U$$. $$L$$ and $$U$$ are triangular, so we just need to compute the product of the diagonals. $$P$$ is not triangular, but if the elements of the diagonal are not 1, they will be zero, and then there has been a swap. So we simply subtract the sum of the diagonal from the length of the diagonal and then subtract 1 to get the number of swaps.

import numpy as np
from scipy.linalg import lu

A = np.array([[6, 2, 3],
[1, 1, 1],
[0, 4, 9]])

P, L, U = lu(A)

nswaps = len(np.diag(P)) - np.sum(np.diag(P)) - 1

detP = (-1)**nswaps
detL =  np.prod(np.diag(L))
detU = np.prod(np.diag(U))

print detP * detL * detU

print np.linalg.det(A)

24.0
24.0


According to the numpy documentation, a method similar to this is used to compute the determinant.