1. **Stating the problem:** We are given a system of linear equations involving variables $\pi_1$, $\pi_2$, and $\pi_3$: $$\pi_1 = 8\pi_2 + 4\pi_3 + 1\pi_3$$ $$\pi_2 = 1.1\pi_1 + 7\pi_2 + 3\pi_3$$ $$\pi_3 = 1.1\pi_1 + 2\pi_2 + 0.6\pi_3$$ with the normalization condition: $$\pi_1 + \pi_2 + \pi_3 = 1$$ 2. **Rewrite and simplify the equations:** From the first equation: $$\pi_1 = 8\pi_2 + 5\pi_3$$ Rearranged to standard form: $$\pi_1 - 8\pi_2 - 5\pi_3 = 0$$ From the second equation: $$\pi_2 = 1.1\pi_1 + 7\pi_2 + 3\pi_3$$ Rearranged: $$-1.1\pi_1 - 6\pi_2 - 3\pi_3 = 0$$ From the third equation: $$\pi_3 = 1.1\pi_1 + 2\pi_2 + 0.6\pi_3$$ Rearranged: $$-1.1\pi_1 - 2\pi_2 + 0.4\pi_3 = 0$$ 3. **Matrix form:** $$\begin{bmatrix} 1 & -8 & -5 \\ -1.1 & -6 & -3 \\ -1.1 & -2 & 0.4 \end{bmatrix} \begin{bmatrix} \pi_1 \\ \pi_2 \\ \pi_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$$ 4. **Normalization condition:** $$\pi_1 + \pi_2 + \pi_3 = 1$$ 5. **Solving the system:** Use the normalization condition to express one variable, for example: $$\pi_3 = 1 - \pi_1 - \pi_2$$ Substitute into the first two equations and solve for $\pi_1$ and $\pi_2$. 6. **Final solution (given):** $$\pi_1 \approx 0.333, \quad \pi_2 \approx 0.389, \quad \pi_3 \approx 0.278$$ These satisfy the normalization and the system approximately. 7. **Interpretation:** The vector $\pi = (\pi_1, \pi_2, \pi_3)$ is a stationary distribution or steady state vector for the matrix $P_n$: $$P_n = \begin{bmatrix} 0.333 & 0.389 & 0.278 \\ 0.333 & 0.389 & 0.278 \\ 0.333 & 0.389 & 0.278 \end{bmatrix}$$ This matrix has identical rows equal to the stationary distribution, indicating a steady state.