Question

Let $A$ be a square matrix such that $A^2=I$, then at least one of $I-A$ and $I+A$ is

• A. singular
• B. symmetric
• C. skew symmetric
• D. none-singular

Answer 1

The correct option is A singular

Given, $A^2=I$
$\Rightarrow I-A^2=(I-A)(I+A)=O$
$\Rightarrow |I-A||I+A|=0$
$\therefore |I-A|=0$ or $|I+A|=0$
i.e atleast one of $(I-A)$ and $(I+A)$ is singular.
Hence, option A.