Exercise 4.3.21

First, if $C$ is not invertible, the set of row vectors of $C$ is not independent. This means the set of row vectors $\left(\begin{array}{cc}\hfill O\hfill & \hfill C\hfill \end{array}\right)$isalso not independent. So it’s impossible that $M$ has $n$ independent rows and hence it’s impossible that $M$ is invertible. The conclusion is that if $C$ is not invertible, we have

Second, if $C$ is invertible, we have the identity

So we get the identity

and hence