Prove that if $ AB^2-A$ is a non singular matrix $BA-A$ is also a non singular matrix. Need help with this problem, tried too many times with failure.

Let $A$ and $B$ be some $n\times n$ matrices. Prove that if$\space$ $ AB^2-A$ $\space$ is a non singular matrix,
$\space$ $BA-A$$\space$  is also a non singular matrix.


Note that $$AB^2-A=A(B^2-I)=A(B-I)(B+I).$$
Since it is nonsingluar, its determinant is not zero. Therefore, 
$$0\neq\det(AB^2-A)=\det A\det(B-I)\det(B+I).$$
In particular, we have $\det A\neq 0$ and $\det(B-I)\neq 0$. Therefore, 
$$\det(BA-A)=\det(B-I)\det A\neq 0$$
which implies that $BA-A$ is nonsingular. 
