Question

State whether the following statement is True or False:
If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.

• A. True
• B. False

Answer 1

The correct option is A True

Determining the transpose of $(A+B+C)$.
Given: $A$, $B$, $C$ are three symmetric matrices of same order.
Let $M=(A+B+C)$
Taking transpose on both sides
Hence, $M^T={(A+B+C)}^T$
$\Rightarrow M^T=(A^T+B^T+C^T)$
$[\because {(A+B)}^T=(A^T+B^T)]$
$[\because A^T=A,B^T=B\text{and}{C}^T=C\text{as}A,B,C\text{are symmetric matrices}]$
$\Rightarrow M^T=(A+B+C)$
$\Rightarrow M^T=M$
$\Rightarrow M\text{is symmetric matrix}$
Hence, the given statement s true.