Set of matrices to the power n^2 are linearly dependent

Question

We are asked to show that the set $S={I_n, A, A^2,...,A^{n^2}}$ is linearly dependent, given $A$ is a square $n$ by $n$ matrix.

I'm not quite sure where to start. We've been doing span, linear combinations, transformations, matrix multiplication, etc. I think A should be invertible, because we must show that:

$a_1 I_n+a_2 A+a_3 A^2+...+a_k A^{n^2}=0$

If the coefficient of the identity element isn't zero, then I can show A must be invertible, since

$a_2 A+a_3 A^2+...+a_k A^{n^2}=-a_1 I_n$

Then

$(x_1 I_n+x_2 A+...+x_k A^{n^2-1}) A= I_n$

No idea where to go from here.

Answer 1

The dimension of the space of $n\times n$ real matrices is only $n^2$, so you can't have a linearly independent set of size $n^2+1$.