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If $A$ and $B$ are two matrices such that $A B=B$ and $B A=A$, then $A^2+B^2=$
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Verified Answer
The correct answer is:
$A+B$
We have $A B=B$ and $B A=A$.
Therefore
$A^2+B^2=A A+B B$ $=A(B A)+B(A B)$
$=(A B) A+(B A) B=B A+A B=A+B$
( $A B=B$ and $B A=A$ )
Therefore
$A^2+B^2=A A+B B$ $=A(B A)+B(A B)$
$=(A B) A+(B A) B=B A+A B=A+B$
( $A B=B$ and $B A=A$ )
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