Let $n\in\mathbb{Z}_{>1}$. Consider the following set : $$ E_n=\{A\in\mathcal{M}_n(\mathbb{C})| \mathrm{Tr}(A)=\det(A)\} $$
My goal is to characterize $E_n$ for any $n$.
I have already found a solution for $n=2$ :
Let A be a matrix in $E_2$ and let's call $\chi_A$ its characteristic polynomial. Since we have $\mathrm{Tr}(A)=\det(A)=\alpha$, then $\chi_A=X^2-\alpha X+\alpha=(X-\lambda_1)(X-\lambda_2)$
With $\lambda_{1/2}=\frac{\alpha\pm\sqrt{\alpha(\alpha-4)}}{2}$
We distinguish four cases :
If $\alpha=0$ then $A$ is similar to $\begin{pmatrix} 0 & a\\ 0 & 0 \end{pmatrix}$
If $\alpha=4$ then $A$ is similar to $\begin{pmatrix} 2 & a\\ 0 & 2 \end{pmatrix}$
Else $A$ is similar to $\begin{pmatrix} \lambda_1 & 0\\ 0 & \lambda_2 \end{pmatrix}$
Do you have any idea on how to proceed for $n=3$ or bigger ?
n.b. This question is entirely from me, I invented it. I do not even know if there is a nice answer to this problem.