Question

If for the non-singular matrix $A, A^{2} = I$, then find $A^{-1}$.

• A. $A$
• B. $I$
• C. $O$
• D. None of these

Answer 1

Answer: (A)

$A$

Answer 2

Given, $A^{2} = I$ Since, A is non-singular matrix $\therefore\quad\left|A\right|\ne0, so, A^{-1}$ exists. Multiplying by $A^{-1}$ on both sides, we get $A^{-1}\left(A^{2}\right)=A^{-1}\left(I\right)$ $\Rightarrow\quad\left(A^{-1}A\right)A=A^{-1} \Rightarrow\, IA=A^{-1} \quad\left(\because\quad A^{-1}A=I\right)$ $\therefore\quad A^{-1}=A \quad\left(\because\quad IA=A\right)$