SEEMOUS 2020 Problem 1

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.