Question
Use the LU decomposition method to solve the following system of equations:
Answer :
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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 & 0 & u_{33} \end{bmatrix}. ] From (A = LU): * (u_{11} = 2) * (u_{12} = 1) * (u_{13} = -1) Next, (l_{21} u_{11} = a_{21} \implies l_{21} = \frac{1}{2} = 0.5) (l_{31} u_{11} = a_{31} \implies l_{31} = \frac{-1}{2} = -0.5) Now (u_{22} = a_{22} - l_{21} u_{12} = 2 - (0.5)(1) = 1.5) (u_{23} = a_{23} - l_{21} u_{13} = 1 - (0.5)(-1) = 1 + 0.5 = 1.5)
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