Question

If AB=A and BA=B, show that A^2=A and B^2=B.
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Answer 1

AB=A

AB(A)=A.A

A(BA)=A^2

Since BA=B

AB=A^2

Similarly

B=B^2

Thus,A^2+B^2=A+B.

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Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

GIVEN

For two matrices A, B it is given that AB=A and BA=B

TO PROVE

${A}^{2} =A \: \: and \: \:{ B}^{2} =B$

PROOF

${A}^{2}$

$= AA$

$= \: (AB)A \: \: \: \: \: ( \because \:AB=A )$

$= \: A(BA ) \: \:$

$= \: AB \: \: \: \: \: ( \because \: BA=B )$

$= \: A\: \: \: \: \: ( \because \:AB=A )$

Again

${B}^{2}$

$= BB$

$= \: (BA)B \: \: \: \: \: ( \because BA=B )$

$= \: B(AB ) \:$

$= \: BA \: \: \: \: \: ( \because AB=A )$

$= B \: \: \: \: \: ( \because BA=B )$

Hence proved