We can write second order systems as a matrix. Any second order system can be written  {jatex options:inline}\mathbf{a}_{ij}{/jatex}  or as a matrix.
{jatex options:inline} \left( \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} \right) {/jatex}
In general the  {jatex options:inline}\mathbf{a}_{ij}{/jatex}  are not related so there are 3{sup}3{/sup =9 components. If the i,j can run from 1 to  {jatex options:inline}n{/jatex}  there are  {jatex options:inline}n^2{/jatex}  components. For a symmetric system  {jatex options:inline}\mathbf{a}_{ij} = a_{ji}{/jatex}
The matrix above becomes   {jatex options:inline} \left( \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{13} & a_{23} & a_{33} \end{array} \right) {/jatex}
There are  {jatex options:inline}\frac{3(3-1)}{2} +3=6{/jatex}  components.
If the i,j can run from 1 to  {jatex options:inline}n{/jatex}  there are  {jatex options:inline}\frac{n(n-1)}{2} +n=\frac{n(n+1)}{2}{/jatex}  components. For a skew symmetric system  {jatex options:inline}\mathbf{a}_{ij} = -a_{ji}{/jatex}
All the diagonal elements must be zero.
The matrix above becomes   {jatex options:inline} \left( \begin{array}{ccc} 0 & a_{12} & a_{13} \\ -a_{12} & 0 & a_{23} \\ -a_{13} & -a_{23} & 0 \end{array} \right) {/jatex}
There are  {jatex options:inline}\frac{3(3-1)}{2}=3{/jatex}  components.
If the i,j can run from 1 to  {jatex options:inline}n{/jatex}  there are  {jatex options:inline}\frac{n(n-1)}{2}{/jatex}  components.