1. **State the problem:** Solve the system of linear equations given by the matrix equation $$\begin{bmatrix}3 & -1 & 0 \\ -1 & 3 & -1 \\ 0 & -1 & 3\end{bmatrix} \begin{bmatrix}p \\ q \\ r\end{bmatrix} = \begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}$$ using the Gauss elimination method. 2. **Write the augmented matrix:** $$\left[\begin{array}{ccc|c} 3 & -1 & 0 & 1 \\ -1 & 3 & -1 & 0 \\ 0 & -1 & 3 & 0 \end{array}\right]$$ 3. **Step 1: Make the first pivot 1 (optional but common):** Divide row 1 by 3: $$R_1 \to \frac{1}{3}R_1 = \left[1, -\frac{1}{3}, 0, \frac{1}{3}\right]$$ 4. **Step 2: Eliminate the first element in rows 2 and 3:** - For row 2: $$R_2 + R_1 = [-1 + 1, 3 - \frac{1}{3}, -1 + 0, 0 + \frac{1}{3}] = [0, \frac{8}{3}, -1, \frac{1}{3}]$$ - Row 3 already has 0 in first column, no change. 5. **Step 3: Make the second pivot 1:** Divide row 2 by $\frac{8}{3}$: $$R_2 \to \frac{3}{8} R_2 = \left[0, 1, -\frac{3}{8}, \frac{1}{8}\right]$$ 6. **Step 4: Eliminate the second element in rows 1 and 3:** - For row 1: $$R_1 + \frac{1}{3} R_2 = \left[1, 0, -\frac{1}{8}, \frac{11}{24}\right]$$ - For row 3: $$R_3 + R_2 = \left[0, 0, \frac{21}{8}, \frac{1}{8}\right]$$ 7. **Step 5: Make the third pivot 1:** Divide row 3 by $\frac{21}{8}$: $$R_3 \to \frac{8}{21} R_3 = \left[0, 0, 1, \frac{1}{21}\right]$$ 8. **Step 6: Eliminate the third element in rows 1 and 2:** - For row 1: $$R_1 + \frac{1}{8} R_3 = \left[1, 0, 0, \frac{2}{3}\right]$$ - For row 2: $$R_2 + \frac{3}{8} R_3 = \left[0, 1, 0, \frac{1}{3}\right]$$ 9. **Final solution:** $$p = \frac{2}{3}, \quad q = \frac{1}{3}, \quad r = \frac{1}{21}$$ This means the vector $$\begin{bmatrix}p \\ q \\ r\end{bmatrix} = \begin{bmatrix}\frac{2}{3} \\ \frac{1}{3} \\ \frac{1}{21}\end{bmatrix}$$ satisfies the original system.