If $A \in M_n(R)$ be diagonalizable with $\operatorname{\mathbf{tr}}(A^2)=0$ show $A$ is the zero matrix
If $\operatorname{tr}(A^2)=0$ then $\sum a_{i,i}^2+\sum2a_{i,j}a_{j,i}=0$
We also have $A^2=CDC^{-1}CDC^{-1}=CD^2C^{-1}$ $$\operatorname{tr}(A^2)=\operatorname{tr}(CD^2C^{-1})$$ $$0=\operatorname{tr}(A^2)=\operatorname{tr}(C^{-1}CD^2) = \operatorname{tr}(D^2)$$
But I'm not sure what to do with this information