Degree $4$ polynomial $p$ such that $p(A) = 0$ for all $A \in M_{2 \times 2}(\mathbb{R})$?

Question

Someone asked me how to prove that there exist real numbers $a_0, \ldots, a_4$ not all zero such that $$ a_0 I_2 + a_1 A + a_2 A^2 + a_3 A^3 + a_4 A^4 = 0 \quad \forall A \in M_{2 \times 2}(\mathbb{R}), $$ where $I_2$ is the identity matrix of dimension $2$. However I don't know how to handle this so I ask for any idea ? The only result I know of that looks like it might be of some help is the Cayley-Hamilton theorem, but here $A$ is general... In fact this result seems a little strange to me because of the generality of $A$. How is this possible ?

Answer 1

There isn't any such 5-tuple. Proof: Choose any $r \in \mathbb R$ and take $A = r I$. Then the equation gives $$ a_0 + a_1 r + a_2 r^2 + a_3 r^3 + a_4 r^4 = 0. $$ Since the LHS is a polynomial in $r$ with infinitely many roots it is the zero polynomial hence $a_0 = \cdots = a_4 = 0$.