### Symmetric matrix definition

A matrix $A$ is symmetric if it is equal to its transpose, i.e., $A=A^T$. A matrix $A$ is symmetric if and only if swapping indices doesn't change its components, i.e., $a_{ij} = a_{ji}$.

The matrix \begin{gather*} A=\left[ \begin{array}{cccc} 1 & 2 & 3 & 4\\ 2 & 5 & 6 & 7\\ 3 & 6 & 8 & 9\\ 4 & 7 & 9 & 10 \end{array} ] \end{gather*} is symmetric while the matrix \begin{gather*} B=\left[ \begin{array}{cccc} 1 & 2 & 3 & 4\\ 12 & 5 & 6 & 7\\ 3 & 6 & 8 & 9\\ 4 & 7 & 9 & 10 \end{array} ] \end{gather*} is not symmetric as $2=a_{12} \ne a_{21}=12$.