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Matrices & Determinants: Involutory Matrix Given Pmatrix Pmatrix Inverse

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

Question

A

is an involutory matrix given by

A = \begin{pmatrix}0 & 1 & -1\\ 4 & -3 & 4\\ 3 & -3 & 4\end{pmatrix}

, then the inverse of

\dfrac{A}{2}

will be:

2A

correct

\dfrac{A^{-1}}{2}

\dfrac{A}{2}

A^2

Solution

Solution Step 1: Use the property of an involutory matrix Since

A

is involutory,

A^2 = I

, which implies

A = A^{-1}

. Step 2: Compute the inverse of

\dfrac{A}{2}

\frac{A}{2}\cdot X = I \implies X = \frac{A}{2}\cdot\left(\frac{A}{2}\right)^{-1}

Testing

X = 2A

\dfrac{A}{2}\cdot 2A = A^2 = I

. Therefore:

\left(\frac{A}{2}\right)^{-1} = 2A

Answer: (1)

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