Question

Let matrix B and A both be nxn symmetric matrices and C be any nxn non-symmetric matrix then \[A=BCB^{T}\].
Then \[A^{T}=(BCB^{T})^{T}\]
\[A^{T}=BC^{T}B^{T}\]
I know that for any matrices, its transpose multiply by itself will give a symmetric matrix. But how come the one above that I have written have this weird behaviour that the middle matrix can be anything and still turns out to be a symmetric matrix?

Answer 1

it is just because (B^T)^T = B ...there you got the simply B

Answer 2

But what's in the middle, the matrix C is any random matrix. Is like how come any matrix C will still bring this whole equation back to a symmetric matrix?