Question

If $A$ is a non-diagonal involutory matrix, then

• A. $A-I=O$
• B. $A+I=O$
• C. $A-I$ is nonzero singular
• D. none of these

Answer 1

Answer: (C)

$A-I$ is nonzero singular

As $A$ is an involutory matrix

$\Rightarrow A^2=I\Rightarrow A^2-I=0\Rightarrow (A-I)(A+I)=0$

Therefore, either $|A+I|=0or|A-I|=0$.

If $|A-I|\ne 0$ Then $(A+I)(A-I)=0\Rightarrow A-I=0$ which is not so.

$\therefore |A-I|=0$ and $A-I\ne O$

Hence $(A-I)$ is nonzero singular