Question:
If $A$ and $B$ are symmetric matrices of the same order, then $A B$ is symmetric iff _________
Solution:
It is given that, A and B are symmetric matrices of the same order.
$\therefore A^{T}=A$ and $B^{T}=B$ ....(1)
Now, $A B$ is symmetric if
$(A B)^{T}=A B$
$\Rightarrow B^{T} A^{T}=A B$
$\Rightarrow B A=A B$ [Using (1)]
Thus, if $A$ and $B$ are symmetric matrices of the same order, then $A B$ is symmetric iff $A B=B A$.
If $A$ and $B$ are symmetric matrices of the same order, then $A B$ is symmetric iff $A B=B A$