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Show that if A and B are square matrices such that AB=BA,then\[(A+B)^2=A^2+2AB+B^2\]

1 Answer

Toolbox:
  • $(A+B)^2=A^2+2AB+B^2$
Step1:
To prove
$(A+B)^2=A^2+2AB+B^2$
LHS:-
$(A+B)^2=(A+B)(A+B)$
$\;\;\;\;\;\qquad=A(A+B)+B(A+B)$
$\;\;\;\;\;\qquad=AA+AB+BA+B.B$
$\;\;\;\;\;\qquad=A^2+AB+BA+B^2$
Step2
Given:
AB=BA.
Hence replace AB=BA.
$\Rightarrow A^2+AB+AB+B^2$
$\Rightarrow A^2+2AB+B^2$
Hence proved.
$(A+B)^2=A^2+2AB+B^2$
answered Mar 23, 2013 by sharmaaparna1
 

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