Olympiad problems

2020 SEEMOUS, 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.

2003 Romania National Olympiad, 4

Tags: matrix , adjugate , polynomial , algebra , linear algebra

Let be a $ 3\times 3 $ real matrix $ A. $ Prove the following statements. [b]a)[/b] $ f(A)\neq O_3, $ for any polynomials $ f\in\mathbb{R} [X] $ whose roots are not real. [b]b)[/b] $ \exists n\in\mathbb{N}\quad \left( A+\text{adj} (A) \right)^{2n} =\left( A \right)^{2n} +\left( \text{adj} (A) \right)^{2n}\iff \text{det} (A)=0 $ [i]Laurențiu Panaitopol[/i]

2010 Laurențiu Panaitopol, Tulcea, 4

Tags: linear algebra , matrix , adjugate , general linear matrix

Let be an odd integer $ n\ge 3 $ and an $ n\times n $ real matrix $ A $ whose determinant is positive and such that $ A+\text{adj} A=2A^{-1} . $ Prove that $ A^{2010} +\text{adj}^{2010} A =2A^{-2010} . $ [i]Lucian Petrescu[/i]