Matrix equation- is this proof legit?

Question

Given that B is a symmetric matrix. C is from the same order as matrix B (C is not necessarily symmetric).

proove that : $CBC^{T}$ is symmetric.I need to proove that $CBC^{T}=(CBC^{T})^{T}$

My way:

$B=B^{T}/*(C)$

$CB=CB^{T}/*(C^{T})$

$CBC^{T}=CB^{T}C^{T}$

$C=(C^{T})^{T}$

$CBC^{T}=(C^{T})^{T}B^{T}C^{T}$

Using $(ABC)^{T}=C^{T}B^{T}A^{T}$

I get : $CBC^{T}=(CBC^{T})^{T}$

I am new to this subject, so I doubt that it is correct. Can someone please tell if this is valid and if not, then what procedure should be used in order to solve such mathematical exercises.

Answer 1

It seems OK.

Shorter: $$ (C B C^T)^T = (C^T)^T B^T C^T = C B C^T $$

Answer 2

Hint: Just use $(AB)^{T} = B^{T}A^{T}$.