Question: Let $A\in M_2(\mathbb R)$, Prove that $a_1A^1 + a_2A^2 + ... + a_5A^5 = 0$ for some $a_i\in\mathbb R$ which are not all zero.
First I figured out the point of this proof is to show $A^1, A^2... A^5$ are linearly dependent.
While since $A^5$ is a 2x2 matrix, there must be a relationship between $A^1$ to $A^5$ such that $A^5$ can be represented by a combination of $A^1,A^2,A^3,A^4$. I can compute that but I think there should be a smarter way to get the dependent we want. Thx.