Exercise 1.4.5

$\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill l\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{cc}\hfill d\hfill & \hfill e\hfill \\ \hfill 0\hfill & \hfill f\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill d\hfill & \hfill e\hfill \\ \hfill \mathit{ld}\hfill & \hfill \mathit{le}+f\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 3\hfill \end{array}\right]$

So we have $d=0$, $e=1$, but also $\mathit{ld}=0$, which doesn’t equal to 2, and contradicts.

$\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill l\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill m\hfill & \hfill n\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill d\hfill & \hfill e\hfill & \hfill g\hfill \\ \hfill 0\hfill & \hfill f\hfill & \hfill h\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill i\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill d\hfill & \hfill e\hfill & \hfill g\hfill \\ \hfill \mathit{ld}\hfill & \hfill \mathit{le}+f\hfill & \hfill \mathit{lg}+h\hfill \\ \hfill \mathit{md}\hfill & \hfill \mathit{me}+\mathit{nf}\hfill & \hfill \mathit{mg}+\mathit{nh}+i\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 1\hfill \end{array}\right]$

We get $d=e=1$, and $g=0$, from $\mathit{ld}=1$, we have $l=1$, and from $\mathit{lg}+h=2$, we have $h=2$, and from $\mathit{le}+f=1$, we have $f=0$, from $\mathit{md}=1$, we have $m=1$, but then $\mathit{me}+\mathit{nf}=1+0=1$ which doesn’t equal to $2$, contradicts.