1. **Problem statement:** Given the linear system: $$\begin{cases} x_1 + x_2 + x_3 = 1 \\ -x_1 + 2x_3 + x_4 = 4 \end{cases}$$ We want to find: (a) The rank of matrix $A$ in $Ax = b$. (b) The rank of the augmented matrix $(A,b)$. (c) The number of solutions of the system. 2. **Matrix form:** The coefficient matrix $A$ and vector $b$ are: $$A = \begin{bmatrix} 1 & 1 & 1 & 0 \\ -1 & 0 & 2 & 1 \end{bmatrix}, \quad b = \begin{bmatrix} 1 \\ 4 \end{bmatrix}$$ 3. **Rank of $A$:** Rank is the number of linearly independent rows (or columns). Row 1: $(1,1,1,0)$ Row 2: $(-1,0,2,1)$ Check if Row 2 is a multiple of Row 1: No scalar $k$ satisfies $k(1,1,1,0) = (-1,0,2,1)$. Therefore, rows are linearly independent. So, $r(A) = 2$. 4. **Rank of augmented matrix $(A,b)$:** Augmented matrix: $$\left[ \begin{array}{cccc|c} 1 & 1 & 1 & 0 & 1 \\ -1 & 0 & 2 & 1 & 4 \end{array} \right]$$ Check if the augmented column changes rank. Since the two rows remain linearly independent including $b$, $r(A,b) = 2$. 5. **Number of solutions:** Number of variables $n=4$, rank $r=2$. Since $r(A) = r(A,b) = 2 < n$, the system has infinitely many solutions. **Final answers:** (a) $r(A) = 2$ (b) $r(A,b) = 2$ (c) Infinitely many solutions