Suppose $V$ is finite-dimensional and $S,T \in \mathcal L(V)$. Prove that $ST = I$ if and only if $TS = I$.

This is a rewording of 3.69, If $ST = I$ then $T$ is clearly injective, which by 3.69 implies $T$ is invertible. Applying $T^{-1}$ to both sides of $ST = I$ shows $S = T^{-1}$ which shows $TS = I$ as desired.

The reverse direction is exactly the same if you swap $T$ and $S$.