Theory

Introduction

Two facts about determinants allow us to use LU factorization to compute the determinant of any matrix:

Two determinant facts

• The determinant of any triangular matrix is equal to the product of its diagonal entries.
• The determinant is multiplicative: $\det AB=\det A\cdot \det B$.

If $A=LU$, then $\det A=\det L\cdot \det U$, and $\det L=1$ assuming it has only ones on the diagonal. So $\det A$ is the product of the diagonal entries of $U$.

If $PA=LU$, then $\det P\cdot \det A=\det L\cdot \det U$. It turns out that $\det P=\pm 1$. So $\det A=\mp \det U$ where again $\det U$ is the product of diagonal entries. The choice of sign is given by $\det P=(-1{)}^k$ where $k$ is the number of non-trivial row-swaps that are composed to give $P$.