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

Show: If A is non-singular then [tex]\alpha_0[/tex] does not equal 0.

So, I get that 0=q(A)=[tex]\alpha_0 I_n + \alpha_1 A + \alpha_2 A^2 +...+A^m[/tex], 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! :)