Let $A$ be a $2\times2$ matrix with real entries such that $Tr(A)=0$ and $detA=-1$.
$1$. Prove that there is a basis of $\mathbb{R}^2$ consisting of eigenvectors of $A$.
$2$. Suppose that $T$ is a $2\times2$ real matrix with respect to the above basis such that $TA=AT$. Prove that $T$ is a diagonal matrix with respect to that basis.
I have answered $1$, that is not too hard as I can calculate the eigenvalues ${-1,1}$ and hence there are two independent eigenvectors of $A$. I'm having problem to prove $2$. Any help, please?
Hint Write $A=PDP^{-1}$. Let $B=PCP^{-1}$ for some matrix $P$.
Show that $CD=DC$, and then simply write $C= \begin{bmatrix} a &b \\ c&d \end{bmatrix}$ and see what $CD=Dc$ means.