If $A$ and $B$ are two matrices such that $A B = B$ and $B A = A$ , then $A^{2} + B^{2} =$

2 A B

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2 B A

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A + B

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A B

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Solution

The correct option is B $A + B$
It is given that $A B = B$ and $B A = A$

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

Consider $A^{2} + B^{2} = A A + B B$
$= B A B A + A B A B$
$= B B A + A A B$ (as $A B = B$ and $B A = A$ )
$= B A + A B$ (as $B A = A$ and $A B = B$ )
$= A + B$ (as $B A = A$ and $A B = B$ )

Therefore, $A^{2} + B^{2} = A + B$
Hence option $C$ is correct.

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Similar questions

A

B

A B = B

B A = A

A^{2} + B^{2}

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

A

B

A B = B

B A = A

A^{2} + B^{2}

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