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:
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?
$\endgroup$1 Answer
$\begingroup$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 $602$.
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