Suppose $A$ is a $m \times m$ matrix. Under what condition(s) does the following finite geometric series hold? Note that $I$ is a $m \times m$ identity matrix.
$$\sum_{i=0}^n A(I - A)^i = I - (I - A)^{n+1}$$
I'm referring to https://mast.queensu.ca/~math211/m211oh/m211oh96.pdf, and it seems that the above would hold if $(I - A)$ is invertible, i.e. $|\lambda_i| < 1$ for each eigenvalue of $A$? Does that imply that $A$ also has to be invertible?
Let $S_n=\sum\limits_{k=0}^n Q^k$ where $Q=(I-A)$. Then $$QS_n=\sum\limits_{k=1}^{n+1} Q^k=-I+S_n+Q^{n+1}$$ Bring $S_n$ to the left part: $$(Q-I)S_n=-I+Q^{n+1}$$ now recalling $Q=I-A$ and multiplying by $(-1)$ we get the desired result.
P.S. No assumptions taken.