There's a linear algebra problem I'm having some trouble with:
Let $A$ and $B$ be square matrices with the dimensions $n\times n$.
Prove or disprove:
- If $A^2 + BA$ is invertible, then $A$ is also invertible.
- If $A^2 + BA$ is not invertible, then $A$ isn't invertible either.
Any help with this would be appreciated. I recognize that if $A^2 + BA$ is invertible then there is a matrix $C$ so that $(A^2 + BA)\cdot C = I$ but beyond that I'm a little lost.