Let $\displaystyle P(x) = a0 + a1x + a2x^2 + ... + anx^n$ be a polynomial of degree n in x with real coefficients. For any mxm matrix A, we define $\displaystyle P(A) = a0Im + a1A + a2A^2 + ... + anA^n$.

Show that, if $\displaystyle P(A) = 0m$ and $\displaystyle a0 not = 0$, then A must be invertible. [Hint: isolate Im from the equation P(A) = 0m]