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If A and B are two matrices such that AB =B and BA = A, then $A^{2} + B^{2}$ is equal to

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2AB

2BA

A + B

answer is C.

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Detailed Solution

We have

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

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