# equitable matrices of order $2$

The most general $2\times 2$ equitable matrix is of the form

 $M=\begin{pmatrix}1&\lambda\\ 1/\lambda&1\end{pmatrix}$

for some $\lambda>0$.

Let us consider the matrix

 $M=\begin{pmatrix}m_{11}&m_{12}\\ m_{21}&m_{22}\end{pmatrix}.$

A necessary and sufficient condition for $M$ to be an equitable matrix is that $m_{11},m_{12},m_{21},m_{22}>0$ and

 $\displaystyle m_{11}=m_{11}m_{11},$ $\displaystyle m_{11}=m_{12}m_{21},$ $\displaystyle m_{12}=m_{11}m_{12},$ $\displaystyle m_{12}=m_{12}m_{22},$ $\displaystyle m_{21}=m_{21}m_{11},$ $\displaystyle m_{21}=m_{22}m_{21},$ $\displaystyle m_{22}=m_{21}m_{12},$ $\displaystyle m_{22}=m_{22}m_{22}.$

It follows that $m_{11}=m_{22}=1$, and $m_{12}=1/m_{21}$.

Title equitable matrices of order $2$ EquitableMatricesOfOrder2 2013-03-22 14:58:30 2013-03-22 14:58:30 matte (1858) matte (1858) 4 matte (1858) Example msc 15-00