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Let $A, B$ be two square matrices such that $A + B = A B .$ Then

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

$A B = B A$

$B = - B A$

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

answer is A.

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Detailed Solution

$A B - A - B = O$ ………….(i)

$\begin{array}{ll} \Rightarrow A B - A - B + I = I \\ \Rightarrow (A - I) (B - I) = I \\ \Rightarrow (B - I) (A - I) = I \\ \Rightarrow B A - A - B + I = I \\ \Rightarrow B A - A - B = O \end{array}$ ………….(ii)

From (i) and (ii), we get $A B = B A .$

Now, $(A + B)^{2} = A^{2} + B A + A B + B^{2}$

$\Rightarrow (A + B)^{2} = A^{2} + 2 A B + B^{2} [∵ A B = B A]$

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