If $\mathrm{A}=\left[\begin{array}{cc}-1& 2\\ 2& 3\end{array}\right] \mathrm{B}=\left[\begin{array}{cc}1& -3\\ -3& 4\end{array}\right]$ verify AB – BA is skew symmetric matrix and AB + BA is symmetric matrix.

here, it can be seen that A and B both are symmetric matrices, since
A=A'              i and,
B=B'             ii
now, 
(A B-B A) '
=(A B) ' - (B A) '
=B 'A'-  A'B'
=B A - A B  (from i and ii) as B '=B and A '=A )
= -(A B - B A)
therefore, -(A B - B A )= (A B- B A) '......iii)
from, iii ) (A B- B A) is a skew- symmetric matrix.

now, 
(AB+BA)'
= (AB) '+(BA) '
= B'A' + A'B'
=BA+AB       (from i and ii as B'=B and A'= A)
=AB+BA
HENCE,
(AB+BA) '=(AB+BA)
So, (AB+BA) is a symmetric matrix.
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