Exercise 3

In the following exercises, use the given $LU$ factorizations to solve the equations $A\xvec = \bvec$.

1. Solve the equation $$A\xvec = \left[\begin{array}{rr} 1 & 0 \\ -2 & 1 \\ \end{array}\right] \left[\begin{array}{rr} 3 & 1 \\ 0 & -2 \\ \end{array}\right]\xvec = \twovec{-3}{0}\text{.}$$

2. Solve the equation $$A\xvec = \left[\begin{array}{rrr} 1 & 0 & 0 \\ -2 & 1 & 0 \\ -1 & 2 & 1 \\ \end{array}\right] \left[\begin{array}{rrr} 2 & 1 & 0 \\ 0 & -1 & 3 \\ 0 & 0 & 1 \\ \end{array}\right]\xvec = \threevec{5}{-5}{7}\text{.}$$