Let A and B be any two 3 × 3 matrices. If Ais symmetric and B is skew symmetric, thenthe matrix AB–BA is

We are given $A^{\mathrm{\prime }}=A,B^{\mathrm{\prime }}=B$

Now        $(AB-BA{)}^{\mathrm{\prime }}=(AB{)}^{\mathrm{\prime }}-(BA{)}^{\mathrm{\prime }}=B^{\mathrm{\prime }}{A}^{\mathrm{\prime }}-A^{\mathrm{\prime }}{B}^{\mathrm{\prime }}$

$=BA-AB=-(AB-BA)$

i.e.  $(AB-BA{)}^{\mathrm{\prime }}=-(AB-BA)$

Hence, AB-BA is a skew-symmetric matrix.