Q. If$\ A$ is a non-diagonal involutory matrix, then
A) $A - I = O$
B) $A + I = O$
C) $A - I$ is non-zero singular
D) None of these
Explanation
(c)
$A^{2} = I$
$\Rightarrow A^{2} - I = O$
$\Rightarrow (A + I)(A - I) = 0$
Therefore, either $|A + I| = 0$ or $|A - I| = 0$. If $|A - I| \neq 0$, then $(A + I)(A - I) = 0 \Rightarrow A - I = 0$ which is not so.
$\therefore\ \ |A - I| = 0$ and $A - I \neq 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.