# Sava's question via email about symmetric matrices

#### Prove It

##### Well-known member
MHB Math Helper
Use the result \displaystyle \begin{align*} \left( M\,N \right) ^T = N^T\,M^T \end{align*} to prove that for any matrix \displaystyle \begin{align*} C \end{align*}, \displaystyle \begin{align*} C^T\,C \end{align*} is a symmetric matrix.
A matrix is symmetric if it is equal to its own transpose, so to show \displaystyle \begin{align*} C^T\,C \end{align*} is symmetric, we need to prove that \displaystyle \begin{align*} \left( C^T\,C \right) ^T = C^T\,C \end{align*}.

\displaystyle \begin{align*} \left( C^T\,C \right) ^T &= C^T\,\left( C^T \right) ^T \textrm{ as } \left( M\,N \right) ^T = N^T\,M^T \\ &= C^T\,C \end{align*}

Since for any matrix \displaystyle \begin{align*} C \end{align*}, \displaystyle \begin{align*} \left( C^T\,C \right) ^T = C^T\,C \end{align*}, that means \displaystyle \begin{align*} C^T\,C \end{align*} is a symmetric matrix.