Exercise 4.3.11

By Exercise 4.2.25 we have $\det <!--\; FUNCTION\; APPLICATION\; -->(-M)={(-1)}^n\det <!--\; FUNCTION\; APPLICATION\; -->(M)$. By Theorem 4.8 we have $\det <!--\; FUNCTION\; APPLICATION\; -->(M^t)=\det <!--\; FUNCTION\; APPLICATION\; -->(M)$. So the conclusion is that

If $n$ is odd, we can conclude that $\det <!--\; FUNCTION\; APPLICATION\; -->(M)=0$ and hence $M$ is not invertible¹ . If $n$ is even, we cannot say anything. For example, the matrix $\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right)$is invertible while the matrix $2\times 2$ zero matrix $O_2$ is not invertible.