Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof / 4 / c / iii

A triangular matrix has determinant equal to the product of its diagonal entries, so $\det L=1$ and $\det U=3\cdot 4\cdot (-11)=-132$. A permutation matrix exchanging two rows has determinant $-1$. Using multiplicativity of the determinant in $PA=LU$ yields $(-1)\det A=-132$. Hence $\det A=132$.