 Solution

Gaussian elimination uses first-column multipliers $-2,2,-3$, second-column multiplier $2$ in the last row, and third-column multiplier $1$ there. Recording these in a unit lower-triangular matrix gives the LU decompositionForward substitution in $Ly=b$ gives $y=(-2,-2,8,1{)}^T$. Back substitution gives $x_4=1$, $-3x_3+2x_4=8$, $x_2+2x_3+2x_4=-2$, and $2x_1-x_2+3x_3+2x_4=-2$. Hence the solution is