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

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

equitable matrices of order 2