Question

If $A$ and $B$ are symmetric matrices of the same order, then which one of the following is $NOT$ true?

• A. $A+B$ is symmetric
• B. $A-B$ is symmetric
• C. $AB+BA$ is symmetric
• D. $AB-BA$ is symmetric

Answer 1

Answer: (D)

Given, $A$ and $B$ are symmetric matrix of same order.
$\Rightarrow A=A^{\prime },B=B^{\prime }$...(i)
(i) $(A+B{)}^{\prime }=A^{\prime }+B^{\prime }=A+B$ [from Eq. (i)]
$\Rightarrow (A+B)$ is symmetric.
(ii) $(A-B{)}^{\prime }=A^{\prime }-B^{\prime }=A-B$ [from Eq. (i) $]$
$\Rightarrow (A-B)$ is symmetric.
(iii) $(AB+BA{)}^{\prime }=(AB{)}^{\prime }+(BA{)}^{\prime }$
$=B^{\prime }{A}^{\prime }+A^{\prime }{B}^{\prime }$
$=BA+AB$ [from Eq.(i)]
$=(AB+BA)$
$\Rightarrow (AB+BA)$ is symmetric.
(iv) $(AB-BA{)}^{\prime }=(AB{)}^{\prime }-(BA{)}^{\prime }$
$=B^{\prime }{A}^{\prime }-A^{\prime }{B}^{\prime }$
$=BA-AB$ [from Eq.(i)]
$=-(AB-BA)$
$\Rightarrow (AB-BA)$ is skew-symmetric matrix.