SEEMOUS 2020

Consider $A\in \mathcal{M}_{2020}(\mathbb{C})$ such that $$(1)\begin{cases} A+A^{\times} =I_{2020}, A\cdot A^{\times} =I_{2020}, \end{cases}$$ where $A^{\times}$ is the adjugate matrix of $A$ , i.e., the matrix whose elements are $a_{ij}=(-1)^{i+j}d_{ji}$ , where $d_{ji}$ is the determinant obtained from $A$ , eliminating the line $j$ and the column $i$ . Find the maximum number of matrices verifying $(1)$ such that any two of them are not similar.

Let $n$ be a positive integer, $k\in \mathbb{C}$ and $A\in \mathcal{M}_n(\mathbb{C})$ such that $\text{Tr } A\neq 0$ and $$\text{rank } A +\text{rank } ((\text{Tr } A) \cdot I_n - kA) =n.$$ Find $\text{rank } A$ .