A solution vector to \( (M-I) \bf{x}=\bf{0} \) is a solution to:
\[
\begin{bmatrix}
-0.3 & 0.1 \\
0.3 & -0.1
\end{bmatrix}
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}=
\begin{bmatrix}
0 \\
0
\end{bmatrix}
\]
As the rows of this matrix are a multiple of one another we really have a single equation in two variables, chose one of them as a parameter and solve for the other.
The steady state is the solution to \(M\bf{x}=\bf{x} \), with \(\sum x_i=1\). Which is a solution to \( (M-I)\bf{x}=0 \) with \( \sum x_i=1\), and so can be generated from the general solution found above to \( (M-I)\bf{x}=0 \).
