Here is my problem:

Let A be a complex n x n matrix with minimal polynomial q(x)=the sum from j=0 to m of $\displaystyle \alpha_j x^j$ where $\displaystyle m\leq n$ and $\displaystyle \alpha_m$ = 1.

Show: If A is non-singular then $\displaystyle \alpha_0$ does not equal 0.

So, I get that 0=q(A)=$\displaystyle \alpha_0 I_n + \alpha_1 A + \alpha_2 A^2 +...+A^m$, but I'm not sure what to do here. I assume we will have to use the fact that A is non-singular, but I'm not sure how. Does it maybe involve multiplying both sides by x on the right side? Any hints would be much appreciated!