For symmetric $A$ and $B$, show that $AB$ is symmetric if $AB=BA$?

Question

I could easily prove this using 2x2 matrices and multiplying them together, but how do you generally prove this and using letters not matrices? (this isn't homework, we haven't even taken symmetry yet I am just exploring)

EDIT: this is my attempt at proving it, I don't know whether it's correct or not.

$(AB)^{T} = B^{T}A^{T}$

And since $A$ and $B$ are symmetric, then $B^{T}A^{T} = BA = AB$ (since $AB = BA$)

doing the same for for $(BA)^{T}$ yields the same result hence showing it is in fact symmetric. It feels right but I am just hesitant about it.

EDIT 2: sorry for the bad phrasing of the question, here's the question literally: "Show that $AB$ is symmetric if and only if $AB = BA$ and both of $A$ and $B$ are symmetric."

Answer 1

I think the question, as expressed in the title, is quite adequately and clearly phrased.

Also, it seems clear to me that our OP Eyad H. is clearly on the right track, but didn't quite follow it far enough.

Here's my work:

With

$A^T = A, \; B^T = B, \tag 1$

$AB = BA \Longrightarrow (AB)^T = B^T A^T = BA = AB; \tag 2$

$(AB)^T = AB \Longrightarrow AB = (AB)^T = B^T A^T = BA. \tag 3$

Combining (2) and (3) we have

$AB = BA \Longleftrightarrow (AB)^T = AB. \tag 4$