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If A, B are two matrices such that AB = A, BA = B, then $$A^2 + B^2$$ =
A + B
A - B
2A + B
A + 2B
AB=A and BA=B
Hence A and B are identity matrices
The square of an identity matrix is the matrix itself
Hence $$A^2=A\ and\ B^2=B$$
So $$A^2+B^2=A+B$$
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