Computational cost of calculating the determinant of an $n \times n$ matrix using LU factorization

Question

How to determine the computational cost associated with calculating determinant of an $n \times n$ matrix using the LU factorization?

Answer 1

With $M(n)$ the times for mutliplying 2 matrices, the factorization LU requires $O(M(n))$ with $M(n)>n^2$. Classical matrix multiplication complexity is $M(n)=O(n^3)$, some efficient way gives $M(n)=O(2^{2.373})$. To get such complexity for LU decomposition it is Coppersmith–Winograd algorithm.

Then to get the determinant of the matrix you have to add $2n$ multiplication.