Let $A,B \in \mathcal{M}_n(\mathbb{R}).$ We consider the function $f:\mathcal{M}_n(\mathbb{C}) \to \mathcal{M}_n(\mathbb{C}),$ defined by $f(Z)=AZ+B\overline{Z},$ $Z \in \mathcal{M}_n(\mathbb{C}),$ where $\overline{Z}$ is the matrix whose entries are the conjugates of the corresponding entries of $Z.$ Prove that the following statements are equivalent: $(1)$ the function $f$ is injective; $(2)$ the function $f$ is surjective; $(3)$ the matrices $A+B$ and $A-B$ are invertible.
Problem
Source: Romanian National Olympiad 2024 - Grade 11 - Problem 3
Tags: linear algebra, matrix, function