If $A$ and $B$ are two $n\times n$ matrices, and given that $B$ is symmetric, then is the matrix $C=\text{trn}(A)BA$ necessarily symmetric?

Question

If $A$ and $B$ are two $n\times n$ matrices, and given that $B$ is symmetric, then is the matrix $C=\text{trn}(A)BA$ necessarily symmetric?

I know that given the symmetric matrices $A$ and $B$, then $AB$ is symmetric if and only if $A$ and $B$ follow commutative property of multiplication, i.e., if $AB = BA$... but is that the case here?

Answer 1

Recall that a matrix is called symmetric if it's equal to its own transpose,ie:$X^{T}=X$

As noted by the other users in the comments, also recall that $(PQR)^T=R^TQ^TP^T$

$$(A^TBC)^T=\cdots=\cdots$$

Can you finish?