I'm not sure if this is the right section to post in but any help would be appreciated since I'm new here.

Suppose V is finite dimenstional and S, T, U are in L(V) where L(V) is the set of linear operators from V to itself and STU = I. Show that T is invertible and that T^-1 = US.

What I tried was saying something along the lines of ST(U) shows ST is invertible with inverse U or S(TU) shows S is invertible with inverse TU, I'm just stuck since T is in the middle and linear maps don't commute, or at least I thought they didn't. This might be different since these are all operators. Any help? Thank you!