જો $A$ અને $B$ એ $3\times3$ શ્રેણિક છે . જો $A$ એ સંમિત અને $B$ એ વિસંમિત શ્રેણિક હોય તો $AB - BA$ એ . . ..
JEE MAIN 2014, Difficult
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Let $A$ be symmetric matrix and $B$ be skew symmetric matrix.
$\therefore {A^T} = A$ and ${B^T} = - B$
Consider
${\left( {AB - BA} \right)^T} = \left( {A{B^T}} \right) - {\left( {BA} \right)^T}$
$ = {B^T}{A^T} - {A^T}{B^T}$
$ = \left( { - B} \right)\left( A \right) - \left( A \right)\left( { - B} \right)$
$ = - BA + AB = AB - BA$
This shows $AB-BA$ is symmetric matrix.
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