If A and B are symmetric is AB=BA true? Question

if we have matrices $A$ and $B$ which are symmetric. Both of these satisfy:
$$  A=A^T$$
Is $AB=BA$ true ?

Attempt to solve

Now if $A$ and $B$ are symmetric then $AB$ most be symmetric too.

$$ AB=(AB)^T=B^TA^T=BA $$

problem is how do you proof that if $A$ and $B$ are symmetric also $AB$ is symmetric ?

If somone could provide some insight on this that would ge highly appreciated.

Thanks,

Tuki
Let $$A =\begin {bmatrix} a &amp; b\\ b &amp; c \end {bmatrix} \quad\mbox{and}\quad B = \begin {bmatrix} d &amp; e\\ e &amp; f \end {bmatrix}$$
be two symmetric matrices. Then 
$$AB-BA =\begin {bmatrix} ad+be &amp; ae + bf\\ bd+ec &amp; be + cf\end {bmatrix} -\begin {bmatrix} ad +be &amp; db + ec\\ ea+bf &amp; eb + fc\end {bmatrix}=\begin {bmatrix} 0 &amp; ?\\ ? &amp; 0\end {bmatrix}.$$
Is the right-hand side necessarily the zero matrix?
If $A$ and $B$ are symmetric matrices then $AB=(BA)^T$, so only if $AB$ is symmetric is it true that $AB=BA$.