If A, B are symmetric matrices of same order, then AB – BA is a - Vidyadip

Question

If A, B are symmetric matrices of same order, then AB – BA is a

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Answer

Given A and B are symmetric matrix of same order
$\Rightarrow$ A = A' .......(i)
$\Rightarrow$ B = B' ......(ii)
So, AB - BA = A'B' - B'A' ...(from eqn (i) and (ii) )
$\Rightarrow$ AB - BA = (BA)' - (AB)' ... ($\because$ (AB)' = B'A')
$\Rightarrow$ AB - BA = (-1) ((AB)' - (BA)') ...(taking -1 common)
$\Rightarrow$ AB - BA = -(AB - BA)' ...($\because$ (A - B)' = A' - B')
Here we see that the relation between (AB - BA) and its transpose i.e. (AB - BA)' is (AB - BA) = - (AB -BA)', this implies that (AB - BA) is a skew symmetric matrix.

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