MathNet 2016-part-04 — 184 problems (EN–VI) Question 13

Let $A, B, C, D \in \mathcal{M}_{n}(\mathbb{C})$, $n \geq 2$ and $k \in \mathbb{R}$ be such that $A C + k B D = I_{n}$ and $A D = B C$. Prove that $C A + k D B = I_{n}$ and $D A = C B$.