I'm incredibly rusty at linear algebra, and in preparation for my course I've been doing some review questions. I've been staring at this one for a half hour and still don't know how to approach it:
"Let A be a square matrix such that $A^3 = 0$. Show that the matrix $I + A + 2A^2$ is invertible and find its inverse."
I'm pretty sure I need to find a relationship between $A^3$ and $I + A + 2A^2$, but I'm not sure how. A matrix is invertible if the determinant is nonzero, and I know how to find the inverse of a matrix, but since this is a more theoretical question I'm not entirely certain how to approach it. Any hints would be much-appreciated :)