Explicit worked example of symmetrizing system of conservation laws

To see an explicit example it would help to write out the system as follows. It is almost symmetric already, because it reads $$ \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & \rho & 0 & 0 & 0 \\ 0 & 0 & \rho & 0 & 0 \\ 0 & 0 & 0 & \rho & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix}z_t + \begin{bmatrix} v_1 & \gamma p & 0 & 0 & 0 \\ 1 & \rho v_1 & 0 & 0 & 0 \\ 0 & 0 & \rho v_1 & 0 & 0 \\ 0 & 0 & 0 & \rho v_1 & 0 \\ 0 & 0 & 0 & 0 & v_1\end{bmatrix}z_{x_1} $$ $$ + \begin{bmatrix} v_2 & 0 & \gamma p & 0 & 0 \\ 0 & \rho v_2 & 0 & 0 & 0 \\ 1 & 0 & \rho v_2 & 0 & 0 \\ 0 & 0 & 0 & \rho v_2 & 0 \\ 0 & 0 & 0 & 0 & v_2\end{bmatrix}z_{x_2} + \begin{bmatrix} v_3 & 0 & 0 & \gamma p & 0 \\ 0 & \rho v_3 & 0 & 0 & 0 \\ 0 & 0 & \rho v_3 & 0 & 0 \\ 1 & 0 & 0 & \rho v_3 & 0 \\ 0 & 0 & 0 & 0 & v_3\end{bmatrix}z_{x_3} = 0. $$ It is only necessary to multiply the first line by $(\gamma p)^{-1}$ to make the coefficient matrices symmetric.