Problem
a) What are the advantages of LU decomposition method over Gaussian elimination? b) Solve the linear system below using the \( L U \) decomposition method. \[ \begin{aligned} x_{1}+x_{2}+3 x_{4} & =8 \\ 2 x_{1}+x_{2}-x_{3}+x_{4} & =7 \\ 3 x_{1}-x_{2}-x_{3}+2 x_{4} & =14 \\ -x_{1}+2 x_{2}+3 x_{3}-x_{4} & =-7 \end{aligned} \]
Solution
Step 1 ：\( \begin{bmatrix} 1 & 1 & 0 & 3 \\ 0 & -1 & -2 & -5 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 12 \end{bmatrix} \)
Step 2 ：\( L = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 3 & -2 & 1 & 0 \\ -1 & 3 & 3 & 1 \end{bmatrix}, U = \begin{bmatrix} 1 & 1 & 0 & 3 \\ 0 & -1 & -2 & -5 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 12 \end{bmatrix} \)
Step 3 ：\( x = \begin{bmatrix} \frac{49}{12} \\ -1 \\ 1 \\ \frac{1}{4} \end{bmatrix} \)
