Word Count : 547

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The system of equations is: 1. (2x_1 + x_2 - x_3 = 2) 2. (x_1 + 2x_2 + x_3 = 8) 3. (-x_1 + x_2 - x_3 = -5) We write it in matrix form (A \mathbf{x} = \mathbf{b}): [ A = \begin{bmatrix} 2 & 1 & -1 \ 1 & 2 & 1 \ -1 & 1 & -1 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 2 \ 8 \ -5 \end{bmatrix}. ] Step 1: LU decomposition We want (A = LU), where (L) is lower triangular with 1s on the diagonal, and (U) is upper triangular. Let [ L = \begin{bmatrix} 1 & 0 & 0 \ l_{21} & 1 & 0 \ l_{31} & l_{32} & 1 \end{bmatrix}, \quad U = \begin{bmatrix} u_{11} & u_{12} & u_{13} \ 0 & u_{22} & u_{23} \ 0 ______ _________ _______ _________ ___ _______ _______ ________ __________ ______ __________.
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