Let \(A = \begin{bmatrix} 1 & 0 & 0 \\ 8 & 2 & 0 \\ -23 & 56 & 3\end{bmatrix}\)
and
\(B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)
What is \(\det(A)\det(B)\)?
The answer is -6.
The matrix \(A\) is lower triangular and so \(\det(A)\) is given
by the product of the diagonal entries, which is 6.
The matrix \(B\) is a permutation matrix corresponding to the
permutation \(\left (\begin{array}{cc} 1 & 2 \\ 2 & 1\end{array}\right)\)
which has a single inversion. So \(\det(B) = -1\).