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Given $BA = A$ and $AB = B$ and $A$ and $B$ are two square matrices, why does $BA +AB = A+B$ ?

Please explain.

ADDED:

The actual problem is it was given that,$BA = A$ and $AB = B$ hence find the value of $A^2 + B^2$.

I simply utilized the fact that if $BA = A$ and $AB = B$ then $A$ and $B$ are idempotent matrices, hence $A^2 + B^2 = A + B$ but the module solution is something more algebraic they break $A^2 + B^2$ and then performs usual substitution after that they showed the result but I don't understand how $BA +AB = A+B$.

  • 2
    This isn't true. Have you misquoted a problem?2011-01-15
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    Erm. Isnt this just a trivial matter of substituting in the given conditions?2011-01-15
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    As you edited, $BA=A$ and $AB=B$, is a given of the problem, not something to be derived. But $BA=A$ doesn't make either one idempotent.2011-01-15
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    @Ross: $A^2=A(BA)=(AB)A=BA=A$.2011-01-15
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    @George: OK thanks2011-01-15
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    After the edit the question has become trivial.2011-01-15
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    @ George Lowther: Oh damn,stupid me.2011-01-15

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This isn't true in general. For example take $A$ as the zero matrix and $B$ as the identity matrix. Then the left hand side is just the zero matrix and the right hand side is the identity matrix, which are not the same.

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    Fixed.2011-01-15
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This is not true, indeed consider the case when $A = 0 $ is the zero matrix and $B= Id$ is the identity, then LHS is the zero matrix but RHS is the identity.