Let's solve **Question 2(a)** step by step: --- ## Step 1: **Identify Admittances and Construct Node Equations** We have a network with three nodes (1, 2, 3) and a reference node (0). The admittances are \(G_1, G_2, G_3, G_4, G_5\), and there is a dependent current source \(gV_2\) between node 2 and ground. **Let's write KCL at each non-reference node (1, 2, and 3):** ### At Node 1: \[ I_1 = (G_1 + G_2)V_1 - G_2 V_2 \] **Explanation:** At node 1, \(G_1\) connects to ground, \(G_2\) connects to node 2. --- ## Step 2: **Continue Node Equations and Assemble Y-Matrix** ### At Node 2: \[ I_2 = -G_2 V_1 + (G_2 + G_3 + G_5)V_2 - G_3 V_3 - gV_2 \] **Explanation:** At node 2, \(G_2\) connects to node 1, \(G_3\) to node 3, \(G_5\) to ground, and the dependent source \(gV_2\) (out of node 2 to ground). ### At Node 3: \[ I_3 = -G_3 V_2 + (G_3 + G_4)V_3 \] **Explanation:** At node 3, \(G_3\) connects to node 2, \(G_4\) to ground. **Now, write these equations in matrix (Y-matrix) form:** \[ \begin{bmatrix} I_1 \\ I_2 \\ I_3 \end{bmatrix} = \begin{bmatrix} G_1 + G_2 & -G_2 & 0 \\ -G_2 & G_2 + G_3 + G_5 - g & -G_3 \\ 0 & -G_3 & G_3 + G_4 \end{bmatrix} \begin{bmatrix} V_1 \\ V_2 \\ V_3 \end{bmatrix} \] --- ### **Final Answer** #### The Y-matrix is: \[ \boxed{ Y = \begin{bmatrix} G_1 + G_2 & -G_2 & 0 \\ -G_2 & G_2 + G_3 + G_5 - g & -G_3 \\ 0 & -G_3 & G_3 + G_4 \end{bmatrix} } \] --- ### **Summary of Steps:** 1. **Wrote node equations using KCL and identified admittances and current sources.** 2. **Arranged node equations in matrix form to construct the Y-matrix.**

Let's address **Question 2(b)** step-by-step: --- ### (b) i) Briefly explain the concept of pivoting as applicable to algebraic equations. **Explanation:** Pivoting is a technique used during the process of solving systems of linear algebraic equations (such as via Gaussian elimination) to improve numerical stability and accuracy. It involves rearranging the equations or variables by selecting the largest (or most suitable) coefficient as the pivot element at each step. This helps to avoid division by very small numbers, which can lead to significant numerical errors. **In essence:** - **Partial pivoting:** Swaps rows to position the largest absolute coefficient in the current pivot position (diagonal element). - **Complete pivoting:** Swaps both rows and columns to position the largest absolute coefficient anywhere in the remaining submatrix. **In summary:** Pivoting improves the robustness of the solution process by choosing the most significant elements as pivots during elimination. --- ### (b) ii) Consider the system: \[ \begin{cases} d x_1 + x_2 + 2 x_3 = 2 \\ 5 x_1 + 2 x_2 + 8 x_3 = 1 \\ 2 x_1 - 4 x_2 + 10 x_3 = 3 \end{cases} \] *Note:* The original equations appear to have some inconsistent or typo-like terms ("x_0" in the second equation). Assuming the intended system is: \[ \begin{cases} d x_1 + x_2 + 2 x_3 = 2 \quad \text{(Equation 1)} \\ 5 x_1 + 2 x_2 + 8 x_3 = 1 \quad \text{(Equation 2)} \\ 2 x_1 - 4 x_2 + 10 x_3 = 3 \quad \text{(Equation 3)} \end{cases} \] However, since the variable \(d\) is present and no numerical value is given, and the question asks about pivoting, we can focus on the coefficients to decide on pivoting. --- ### **Step-by-step indication of pivoting:** **Step 1: Write the coefficient matrix** \[ A = \begin{bmatrix} d & 1 & 2 \\ 5 & 2 & 8 \\ 2 & -4 & 10 \end{bmatrix} \] **Step 2: Identify the largest absolute coefficient in the first column (for the first pivot):** - Absolute values: \(|d|\), 5, 2 - To perform partial pivoting, compare these: - If \(|d| < 5\), swap row 1 and row 2 to bring the largest (5) to the pivot position (row 1). **Assuming \(d\) is such that \(|d| < 5\), then pivoting involves:** - **Swapping row 1 and row 2:** \[ \text{After pivoting:} \] \[ \begin{bmatrix} 5 & 2 & 8 \\ d & 1 & 2 \\ 2 & -4 & 10 \end{bmatrix} \] with the corresponding right-hand side (if provided). **Note:** If \(d\) is known and larger than 5, then no pivoting is needed; otherwise, swap to improve stability. --- ### **Final summary:** - **Pivoting** involves selecting the largest magnitude element in the current column as the pivot to reduce numerical errors. - In this system, pivoting would mean swapping rows to position the coefficient with the largest absolute value in the pivot position during elimination. --- **Final answer:** **(b) i)** Pivoting is a technique in solving algebraic equations where row (or column) exchanges are performed to position the largest available coefficient as the pivot element in each step of elimination, enhancing numerical stability. **(b) ii)** For the given system, pivoting involves comparing the first-column coefficients (\(d\), 5, 2). If \(|d| < 5\), swap row 1 and row 2 to bring the coefficient 5 into the pivot position, thereby improving the accuracy of the elimination process.