LU Decomposition Determinant Mismatch Matlab

Question

I'm trying to get the determinant of a matrix by LU factorization.

I have the following matrix:

a = [2 4 2;
    1 5 2;
    4 -1 9];

When I execute the command det(a) in matlab, it shows the determinant to be 48. Then I enter the decomposition command:

[L, U, P] = lu(a)

It shows the matrix L to be:

1.0000         0         0
0.2500    1.0000         0
0.5000    0.8571    1.0000

and the matrix U to be:

4.0000   -1.0000    9.0000
     0    5.2500   -0.2500
     0         0   -2.2857

As we know that the determinant of a matrix is also the product of the principal diagonal elements of it's U matrix after the decomposition, it doesn't match up in this case. Because the product of the diagonal elements of the matrix U is -47.999. But it shouldn't have been a negative number.

What am I missing here?

Answer 1

$$LU=PA$$

$$\det(L) \det(U) = \det(P) \det(A)$$

$$\det(A) = \det(P) \det(U)$$

since $\det(L)=1$.

$\det(P)=-1$ for this example.