If $A$ and $B$ are symmetric matrices of the same order, then which of the following is always true?

Solution

1. Given $A^T=A$ and $B^T=B$.
2. Consider the transpose of the sum: $(A+B{)}^T=A^T+B^T$.
3. Substitute the symmetric properties: $A^T+B^T=A+B$.
4. Since $(A+B{)}^T=A+B$, the matrix $A+B$ is symmetric.
5. Hence the answer is (B).