Consider a system of linear equations \( P X = Q \) where \( P \in \mathbb{R}^{3 \times 3} \) and \( Q \in \mathbb{R}^{3 \times 1} \). Suppose \( P \) has an LU decomposition, \( P = LU \), where:
Which of the following statement(s) is/are TRUE?
Consider solving the following system of simultaneous equations using LU decomposition. \[ x_1 + x_2 - 2x_3 = 4 \\ x_1 + 3x_2 - x_3 = 7 \\ 2x_1 + x_2 - 5x_3 = 7 \] where \( L \) and \( U \) are denoted as \[ L = \begin{pmatrix} L_{11} & 0 & 0 \\ L_{21} & L_{22} & 0 \\ L_{31} & L_{32} & L_{33} \end{pmatrix}, U = \begin{pmatrix} U_{11} & U_{12} & U_{13} \\ 0 & U_{22} & U_{23} \\ 0 & 0 & U_{33} \end{pmatrix} \] Which one of the following is the correct combination of values for \( L_{32}, U_{33}, \) and \( x_1 \)?
