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Matrices & Determinants: Involutory Matrix Given Pmatrix Pmatrix Inverse

JEE Maths question with a full step-by-step solution.

Question
AA is an involutory matrix given by A=(01−14−343−34)A = \begin{pmatrix}0 & 1 & -1\\ 4 & -3 & 4\\ 3 & -3 & 4\end{pmatrix}, then the inverse of A2\dfrac{A}{2} will be:
A2A2Acorrect
BA−12\dfrac{A^{-1}}{2}
CA2\dfrac{A}{2}
DA2A^2
Solution
Solution Step 1: Use the property of an involutory matrix Since AA is involutory, A2=IA^2 = I, which implies A=A−1A = A^{-1}. Step 2: Compute the inverse of A2\dfrac{A}{2}
A2⋅X=I  ⟹  X=A2⋅(A2)−1\frac{A}{2}\cdot X = I \implies X = \frac{A}{2}\cdot\left(\frac{A}{2}\right)^{-1}
 Testing X=2AX = 2A: A2⋅2A=A2=I\dfrac{A}{2}\cdot 2A = A^2 = I. Therefore:
(A2)−1=2A\left(\frac{A}{2}\right)^{-1} = 2A
 Answer: (1)
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