Let $A$ and $Q$ be linear operators on finite dimensional vector space. We know that Q is nilpotent and it commutes with A. Prove that operator $(A + Q)^{-1}$ is invertible if and only if $A$ is invertible.
What I have done so far:
Let suppose A is invertible, than:
$$(A + Q)^{-1} = \frac{1}{A+Q} = \frac{A^{-1}}{1+A^{-1}Q} = A^{-1} \Bigl(\frac{1}{1-(-A^{-1}Q)}\Bigr)$$
Which I re-wrote as:
$$A^{-1}\bigl(1 + (-A^{-1}Q) + (-1A^{-1}Q)^2...\bigr)$$
Now since Q is nilpotent, it means there exist such $m$, such that $Q^m = 0$.
Which means, that from some term onward, all terms are zero so I have: $$A^{-1}\bigl(1 -A^{-1}Q + A^{-2}Q^2... A^{-m}Q^m(-1)^m \bigr)$$
Which shows $(A+Q)^{-1}$ is invertible.
My question is, how do I prove ekvivalence in other direction?
If $A + Q$ is invertible then $-Q$ is nilpotent and commutes with $A + Q$ so $(A + Q) + (-Q)$ is invertible.