Find the $LU$ factorization of $$A= \left[\begin{array}{ccc} 3 &5 &-3\cr -9 &-13 &8\cr 3 &-1 &1 \end{array}\right]$$ such that the main diagonal of \left[\begin{array}{ccc} 1 &0 &0\cr -3 &1 &0\cr 1 &-3 &1 \end{array}\right]$ has all ones and use it to solve the system $$\left[\begin{array}{ccc} 3 &5 &-3\cr -9 &-13 &8\cr 3 &-1 &1 \end{array}\right] \left[\begin{array}{c} x_{1}\cr x_{2}\cr x_{3}\cr \end{array}\right] = \left[\begin{array}{c} -2\cr 7\cr -2 \end{array}\right].$$