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

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$A + B$

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

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

answer is D.

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Detailed Solution

$B = - A^{- 1} B A$

$A B = - A (A^{- 1} . B A) = - (A A^{- 1}) (B A) = - I (B A) = - B A$

$\begin{array}{l} A B = - B A \\ {(A + B)}^{2} = A^{2} + A B + B A + B^{2} \end{array}$

$\begin{array}{l} = A^{2} - B A + B A + B^{2} \\ = A^{2} + B^{2} \end{array}$

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