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To solve the given system of linear equations using Gaussian elimination with partial pivoting, we'll follow these steps: 1. Write the augmented matrix \( [A|b] \). 2. Perform partial pivoting to ensure the largest absolute value in the current column is the pivot. 3. Eliminate to form an upper triangular matrix. 4. Back-substitute to find the solution. 5. Store the multipliers used during elimination. 6. Write the pivoting vectors indicating the row swaps. ### Given System: \[ \begin{aligned} x_1 - x_2 + 3x_3 &= 3 \\ 2x_1 + x_2 + 4x_3 &= 7 \\ 3x_1 + 5x_2 - 2x_3 &= 6. \end{aligned} \] ### Step 1: Augmented Matrix \[ [A|b] = \begin{bmatrix} 1 & -1 & 3 & | & 3 \\ 2 & 1 & 4 & | & 7 \\ 3 & 5 & -2 & | & 6 \end{bmatrix} \] ### Step 2: Partial Pivoting and Elimination #### First Pivot: - The largest absolute value in the first column is 3 (in row 3). - Swap __________ _______ _______ ________ ______ _____ _____ ___ ___.
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