Proving a matrix is invertible

Question

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:

1. If $A^2 + BA$ is invertible, then $A$ is also invertible.
2. 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.

Answer 1

1. $I=C(A^2 + BA)=C(A+B)A$ and so $C(A+B)$ is the inverse of $A$.

2. This is false. Take $A=I$ (or any invertible matrix) and $B=-A$.