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Algebra Level 3

If A \color{#D61F06}{A} and B \color{#3D99F6}{B} are two non-singular matrices such that A B \color{#D61F06}{A}\color{#3D99F6}{B} = B \color{#3D99F6}{B} and B A \color{#3D99F6}{B}\color{#D61F06}{A} = A \color{#3D99F6}{A} ,then.

A 2 + B 2 = \large \color{#D61F06}{A^2} + \color{#3D99F6}{B^2} =

B A \color{#3D99F6}{B} \color{#D61F06}{A} A + B \color{#D61F06}{A} + \color{#3D99F6}{B} 2 A B 2\color{#D61F06}{A} \color{#3D99F6}{B} 2 B A 2\color{#3D99F6}{B} \color{#D61F06}{A} A B \color{#D61F06}{A} \color{#3D99F6}{B}

Akhil Bansal by Akhil Bansal , 22, India

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1 solution

Akhil Bansal
Sep 13, 2015

= A 2 + B 2 = \color{#D61F06}{A^2} + \color{#3D99F6}{B^2} = A ( B A ) + B ( A B ) = \color{#D61F06}{A}(\color{#3D99F6}{B} \color{#D61F06}{A}) + \color{#3D99F6}{B}(\color{#D61F06}{A} \color{#3D99F6}{B}) = ( A B ) A + ( B A ) B = (\color{#D61F06}{A} \color{#3D99F6}{B}) \color{#D61F06}{A} + (\color{#3D99F6}{B} \color{#D61F06}{A}) \color{#3D99F6}{B} = B A + A B = \color{#3D99F6}{B}\color{#D61F06}{A} + \color{#D61F06}{A}\color{#3D99F6}{B} = A + B = \boxed{\color{#D61F06}{A} + \color{#3D99F6}{B}}

7 Helpful 1 Interesting 0 Brilliant 0 Confused

Nice solution ,,

Upvoted.

Syed Baqir - 5 years, 9 months ago

A & B are Idempotent Matrix.

Krishna Shankar - 5 years, 5 months ago

1 pending report

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