Chapter 2.4, Problem 9E

Suppose A₁₁ is an invertible matrix. Find matrices X and Y such that the product below has the form indicated. Also, compute B₂₂. [Hint: Compute the product on the left, and set it equal to the right side.]

$\left[\begin{array}{ccc}I& 0& 0\\ X& I& 0\\ Y& 0& I\end{array}\right]\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\\ A_{31}& A_{32}\end{array}\right]=\left[\begin{array}{cc}{B}_{11}& B_{12}\\ 0& B_{22}\\ 0& B_{32}\end{array}\right]$