Suppose $V$ is finite-dimensional and $S,T \in \mathcal L(V)$. Prove that $ST$ is invertible if and only if both $S$ and $T$ are invertible.

If both $S$ and $T$ are invertible, then $ST$ is obviously invertible with inverse $T^{-1}S^{-1}$.

Now suppose $ST$ is invertible with inverse $R$, then $(RS)T = I$ implies $T$ is injective (it has a left inverse) which by 3.69 means $T$ is invertible.
Since $T$ is invertible and $S = S(TT^{-1}) = (ST)T^{-1}$ is the product of two invertible transformations $S$ is also invertible.