Problem 1

Part of 2020 SEEMOUS

Problems(1)

A Matrix and Its Adjugate

Source: SEEMOUS 2020 P1

A\in \mathcal{M}_{2020}(\mathbb{C})

(1)\begin{cases} A+A^{\times} =I_{2020},\\ A\cdot A^{\times} =I_{2020},\\ \end{cases}

A^{\times}

is the adjugate matrix of

A

, i.e., the matrix whose elements are

a_{ij}=(-1)^{i+j}d_{ji}

d_{ji}

is the determinant obtained from

A

, eliminating the line

j

i

. Find the maximum number of matrices verifying

(1)

such that any two of them are not similar.

linear algebramatrixadjugate