I’m stuck solving this problem. Given $T^n=0$, I know that the column vectors of T will be linearly dependent. However, it’s unclear that subtracting a linearly dependent matrix from the identity gives another linearly independent (invertible) matrix. How do I begin to show this?
The problem continues: show that $(I-T)^{-1} =I+T+...+T^{n-1}$ and guess how one would give the formula. Perhaps that leap of intuition will occur once I understand the first half better, but a hint here would be appreciated as well.