Finding determinant of 4*4 Matrix via LU Decomposition? What is the shortcut way of finding the determinant of a 4 by 4 matrix (and I assume this applies to any n by n square matrix greater than 2) once you have found an LU or PLU decomposition?
Given that det(A)=det(P)det(L)det(U) or det(A)=det(L)det(U) (if no permutation matrix was necessary).
Here is a PLU decomposition example:
http://www.wolframalpha.com/input/?i=LU+decomposition+of+%7B%7B4%2C3%2C2%2C1%7D%2C%7B1%2C10%2C3%2C4%7D%2C%7B5%2C3%2C2%2C-4%7D%2C%7B4%2C8%2C7%2C9%7D%7D
The determinant of that matrix is 602. If I recall, one is supposed to multiply the diagonal entries of the Upper Matrix starting from row 1, column 1 (which gets me -602) and then...    magic? What happens from there?

You seem to have the right ideas.  Here's the gist:


*

*Any permutation matrix has determinant $\pm 1$, depending on the parity of the permutation

*To find the determinant of an upper triangular or lower triangular matrix, take the product of the diagonal entries.

*If $A = PLU$, then $\det(A) = \det(P)\det(L)\det(U)$


In your example, $P$ has determinant $-1$, $L$ has determinant $1$ (since each diagonal entry is $1$), and $U$ has determinant $-602$, as you indicated. So, the determinant of the product is the product of the determinants which is $602$.
