MathNet 2016-part-04 — 184 problems (EN–VI) Question 6

Consider matrices $A$, $B$, $C$, $D \in \mathcal{M}_n(\mathbb{C})$, $n \ge 2$ and $k \in \mathbb{R}$ so that $AC + kBD = I_n$ and $AD = BC$. Prove that $CA + kDB = I_n$ and $DA = CB$.