Suppose $V$ is finite-dimensional and $S,T,U \in \mathcal L(V)$ and $STU = I$. Show that $T$ is invertible and that $T^{-1} = US$.

Clearly $\text{null }U = \{0\}$ meaning $U$ is invertible by 3.69. It's also clear that $\text{range }S = V$ since for each $v \in V$, $S(TUv) = v$ meaning (again by 3.69) $S$ is invertible.
Inverting both of them gives $T = S^{-1}U^{-1}$ which implies $T$ is invertible since it's the product of invertible maps, inverting both sides gives $T^{-1} = US$ as desired.