Exercise 2.7.36

Certainly $B^T$ is northwest. $B^2$ is a full matrix! $B^{-1}$ is southeast: ${\left[\begin{array}{cc}1\hfill & 1\hfill \\ 1\hfill & 0\hfill \end{array}\right]}^{-1}=\left[\begin{array}{cc}\hfill 0& \hfill 1\\ \hfill 1& \hfill -1\end{array}\right]$. The rows of $B$ are in reverse order from a lower triangular $L$, so $B=\mathit{PL}$. Then $B^{-1}=L^{-1}{P}^{-1}$ has the columns in reverse order from $L^{-1}.$ So $B^{-1}$ is southeast. Northwest $B=\mathit{PL}$ times southeast $\mathit{PU}$ is $(\mathit{PLP})U=$ upper triangular.