If $A$ and $B$ are two square matrices such that $B = - A^{- 1} B A,$ then ${(A + B)}^{2} =$

0

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A^{2} + B^{2}

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A^{2} + 2 A B + B^{2}

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A + B

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Solution

The correct option is C $A^{2} + B^{2}$
Given, $B = - A^{- 1} B A$
$∴ A B = - A A^{- 1} B A = - I B A = - B A$
$∴ A B = - B A$
Now ${(A + B)}^{2} = (A + B) (A + B) = A^{2} + A B + B A + B^{2}$
$= A^{2} + B^{2}$ ........ $[∵ B A = - A B]$
Thus, ${(A + B)}^{2} = A^{2} + B^{2}$

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