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

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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 A $A^{2} + B^{2}$
Pre multiply by A
$A B = - A A^{- 1} B A$
$∴ A B = - B A$
$∴ (A + B)^{2} = A^{2} + B^{2} + A B + B A$
$= A^{2} + B^{2}$

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B = - A^{- 1} B A

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B = - A^{- 1} B A

(A + B)^{2} = A^{2} + B^{2}

Show that if A and B are square matrices such that AB = BA, then $(A + B)^{2} = A^{2} + 2 A B + B^{2}$ .

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