Question

Choose the correct answer in the following question:

If A,B are symmetric matrices of same order, then AB-BA is a
(a)skew-symmetric matrix
(b)symmetric matrix
(c)zero matrix
(d)identity matrix

Answer 1

Given, A and B are symmetric matrices.
$\Rightarrow A^{\prime }=Aand{B}^{\prime }=B(AB-BA{)}^{\prime }=(AB{)}^{\prime }-\left(BA{)}^{\prime }\right[\because (A-B{)}^{\prime }=A^{\prime }-B^{\prime }]=B^{\prime }{A}^{\prime }-A^{\prime }{B}^{\prime }[\because (AB{)}^{\prime }=B^{\prime }{A}^{\prime }]=BA-AB[\because A^{\prime }=Aand{B}^{\prime }=B]=-(AB-BA)\Rightarrow (AB-BA{)}^{\prime }=-(AB-BA)$
Thus, (AB-BA)~is a skew -symmetric matrix. So, correct option is (a).