If $A$ and $B$ are matrices and $B = A B A^{- 1}$ then the value of $(A + B) (A - B)$ is

A^{2} + B^{2}

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A^{2} - B^{2}

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A + B

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A - B

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Solution

The correct option is B $A^{2} - B^{2}$
$B = A B A^{- 1}$ (Given)
But $B = B A A^{- 1}$
$∴ A B A^{- 1} = B A A^{- 1} ⟹ A B = B A$
Now $(A + B) (A - B) = A^{2} - A B + B A - B^{2}$
$= A^{2} - A B + A B - B^{2}$ $[∵ A B = B A]$
$= A^{2} - B^{2}$
$∴$ $(A + B) (A - B) = A^{2} - B^{2}$

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