equitable matrices of order $2$

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

for some $\lambda >0$.

Let us consider the matrix

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

$m_{11}=m_{11}{m}_{11},$

$m_{11}=m_{12}{m}_{21},$

$m_{12}=m_{11}{m}_{12},$

$m_{12}=m_{12}{m}_{22},$

$m_{21}=m_{21}{m}_{11},$

$m_{21}=m_{22}{m}_{21},$

$m_{22}=m_{21}{m}_{12},$

$m_{22}=m_{22}{m}_{22}.$

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