Question 61
Let $n \in \mathbb{N}$, $(n \ge 2)$ be a positive integer, and let $A, B \in \mathcal{M}_n(\mathbb{R})$ be two matrices with real entries. Prove that there exists a complex number $z$, with $|z| = 1$, such that
$$
\operatorname{Re}(\det(A + zB)) \ge \det(A) + \det(B),
$$
where we denote by $\operatorname{Re}(w)$ the real part of the complex number $w$.
$$
\operatorname{Re}(\det(A + zB)) \ge \det(A) + \det(B),
$$
where we denote by $\operatorname{Re}(w)$ the real part of the complex number $w$.