Question

If A is a non-diagonal involutory matrix, then

• A. A - I = 0
• B. A + I = 0
• C. A - I is non zero singular
• D. none of these

Answer 1

The correct option is C A - I is non zero singular

$A^2=I\Rightarrow A^2-I=0$
$\Rightarrow$ (A+I)(A-I)=0
$\therefore$ either $|A+I|=0$ or
$|A-I|=0$
If $|A-I|\ne 0$, then $(A+I)(A-I)=0\Rightarrow A+I=0$ which is not so
$\therefore |A-I|$ and $A-I\ne 0$.