(Solved): 5. Consider the \( 5 \times 5 \) matrix system \( A \vec{x}=\vec{b} \) below. \[ \left[\begin{arra ...

5. Consider the \( 5 \times 5 \) matrix system \( A \vec{x}=\vec{b} \) below. \[ \left[\begin{array}{ccccc} 1 & 3 & 1 & -3 & 0 \\ 2 & 8 & 5 & 10 & 1 \\ 3 & 11 & 7 & 10 & 1 \\ 1 & 7 & 7 & 11 & 0 \\ 1 & -1 & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \\ t \\ k \end{array}\right]=\left[\begin{array}{l} 1 \\ 0 \\ 1 \\ 0 \\ 1 \end{array}\right] \] The matrix \( A \) can be factored as \( A=L U=\left[\begin{array}{ccccc}1 & 0 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 & 0 \\ 3 & 1 & 1 & 0 & 0 \\ 1 & 2 & 0 & 1 & 0 \\ 1 & -2 & 5 & -\frac{10}{9} & 1\end{array}\right]\left[\begin{array}{ccccc}1 & 3 & 1 & -3 & 0 \\ 0 & 2 & 3 & 16 & 1 \\ 0 & 0 & 1 & 3 & 0 \\ 0 & 0 & 0 & -18 & -2 \\ 0 & 0 & 0 & 0 & \frac{7}{9}\end{array}\right] \) (a) Factor \( A \) as \( L D U \), where \( L \) and \( U \) are lower and upper unitriangular matrices (triangular matrices with 1 along the diagonal) (b) Solve the system \( A \vec{x}=\vec{b} \) in the following order: 1. Let \( \vec{c}=D U \vec{x} \) and solve for \( \vec{c} \) in \( L \vec{c}=\vec{b} \) using substitution 2. Find \( D^{-1} \). What is special about \( D^{-1} \) and its relationship with \( D \) ? 3. Having found \( \vec{c} \) in part (a), solve \( D U \vec{x}=\vec{c} \) by first multiplying both sides by \( D^{-1} \) and substituting to solve for \( \vec{x} \) (i.e. \( U \vec{x}=D^{-1} \vec{c} \) )