LU Decomposition: Calculating Determinant of Square Matrix (Example Included)

[demipaul]

How would I calculate the determinant of a square matrix using LU Decomposition. Please be plain, I am not good with technical terms. An example would be nice. Thank you!

 

[jeff1evesque]

Hello,

I looked up a nice example on how to compute LU Decomposition. On this http://www.utdallas.edu/dept/abp/PDF_Files/LinearAlgebra_Folder/LU_Decomposition.pdf" , the author has provided a nice clear explanation on how to do an LU decomposition (I've also attached the pdf, in case the link is taken down).

And according to http://en.wikipedia.org/wiki/LU_decomposition#Determinant"

The matrices L and U can be used to compute the determinant of the matrix A very quickly, because det(A) = det(L) det(U) and the determinant of a triangular matrix is simply the product of its diagonal entries. In particular, if L is a unit triangular matrix, then
$$det(A) = det(L)det(U) = 1 \cdot det(U) =\prod^{n}_{i = 1}u_{ii}.$$

(Note: A http://planetmath.org/encyclopedia/UnitLowerTriangularMatrix.html" is a triangular matrix with 1's along it's diagonals)

So if you used the methodology to get an LU decomposition and further reduce the L matrix to a lower triangular matrix with 1's along the diagonal- then you can simply take the product of the diagonals of the right upper triangular matrix U to get your determinant for matrix A.