Chapter 3, Problem 15SE

Let A, B, C, and D be n × n matrices with A invertible.

a. Find matrices X and Y to produce the block LU factorization

$\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]=\left[\begin{array}{cc}I& 0\\ X& I\end{array}\right]\left[\begin{array}{cc}A& B\\ 0& Y\end{array}\right]$ and then show that det $\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]$ = (det A) · det(D − CA⁻¹B)