How to show that B is not invertible

Question

This is not a homework, this question came in my exam, and I did not know how to solve it.

"Let $A$ and $B$ be $3\times3$ matrices such that $AB = - BA$. Show that if $A$ is invertible, then $B$ is not invertible"

I tried starting with the expression $ AB = - BA$ and multiply $A$ by its inverse, but no luck.

Thank you!

Answer 1

Here is the answer:

$$\operatorname{det} (AB) = \operatorname{det} (-BA)\\ \operatorname{det} A \operatorname{det} B = (-1)^3 \operatorname{det} A \operatorname{det} B $$

dividing both equations by $\operatorname{det} A$ ($\operatorname{det} A$ is different than zero since $A$ is invertible)

$$\operatorname{det} B = (-1)^3 \operatorname{det} B$$

$\operatorname{det} B$ must be equal to zero, thus B is not invertible.

Answer 2

We have $ABA^{-1}=-B$, meaning that $B$ and $-B$ have the same eigenvalues. It follows that the eigenvalue of $B$ is 0. Note that the size of the matrices is irrelevant here.