Matrices $A$ and $B$ satisfy $A B = B^{- 1},$ then the matrix $X$ satisfying $A^{- 1} X A = B$ is equal to

A

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B

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I

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A^{- 1}

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Solution

The correct option is B $B$
Given: $A B = B^{- 1}$
$\Rightarrow A B^{2} = I \dots (i)$

Now,
$A^{- 1} X A = B$
$\Rightarrow A A^{- 1} X A = A B$
$\Rightarrow X A B = A B^{2}$
$\Rightarrow X A B = I \dots (Using (i))$
$\Rightarrow X A B^{2} = B$
$\Rightarrow X I = B \dots (Using (i))$
$\Rightarrow X = B$

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