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.
$\Rightarrow A=A',\, B=B'$...(i)
(i) $(A+B)'=A'+B'=A+B$ [from Eq. (i)]
$\Rightarrow (A+B)$ is symmetric.
(ii) $(A-B)'=A'-B'=A-B$ [from Eq. (i) $]$
$\Rightarrow (A-B)$ is symmetric.
(iii) $(AB+BA)'=(AB)'+(BA)'$
$=B'A'+A'B'$
$=BA+AB$ [from Eq.(i)]
$=(AB+BA)$
$\Rightarrow (A B+B A)$ is symmetric.
(iv) $(A B-BA)'=(A B)'-(B A)'$
$=B' A'-A' B'$
$=B A-A B$ [from Eq.(i)]
$=-(A B-B A)$
$\Rightarrow (A B-B A)$ is skew-symmetric matrix.