Exercise 1.4.11

$A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 3\hfill \\ \hfill 2\hfill & \hfill 4\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \frac{3}{4}\hfill \\ \hfill 1\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 2\hfill & \hfill 4\hfill \end{array}\right]+\left[\begin{array}{cc}\hfill -\frac{1}{2}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill \frac{3}{4}\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 2\hfill & \hfill 4\hfill \\ \hfill -\frac{1}{2}\hfill & \hfill 0\hfill \end{array}\right]$

$P_{1}A{P}_2=\mathit{LU}=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill \frac{3}{4}\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{cc}\hfill 4\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill -\frac{1}{2}\hfill \end{array}\right]$