## 每週問題 January 6, 2014

Let $A$ and $B$ be $n\times n$ matrices. If $I-AB$ is nonsingular, show that $I-BA$ is nonsingular.

\displaystyle\begin{aligned} (I-BA)(I+B(I-AB)^{-1}A)&=I-BA+(B-BAB)(I-AB)^{-1}A\\ &=I-BA+B(I-AB)(I-AB)^{-1}A\\ &=I-BA+BA=I.\end{aligned}