Question

If $A$ and $B$ are symmetric matrices of the same order, then $A{B}^{\prime }-B{A}^{\prime }$ is a

• A. skew-symmetric matrix
• B. null matrix
• C. symmetric matrix
• D. None of the above

Answer 1

Answer: (A)

$\left(A{B}^{\prime }-B{A}^{\prime }\right)=\left(A{B}^{\prime }\right)-\left(B{A}^{\prime }\right)$
$=\left(B{A}^{\prime }-A{B}^{\prime }\right)$
$=-\left(A{B}^{\prime }-B{A}^{\prime }\right)$
$\because \left(A{B}^{\prime }-B{A}^{\prime }\right)=-\left(A{B}^{\prime }-B{A}^{\prime }\right)$
Thus, $A{B}^{\prime }-B{A}^{\prime }$ is a skew-symmetric matrix.