$2x+3y+z=-1 \ 5x+y+z=9, \ 3x+2y+4z=11$ The augmented co-efficient matrix can be represented as $A=\begin{bmatrix} a_{11} &a_{12} &a_{13} &a_{14} \a_{21} &a_{22} &a_{23} &a_{24} \a_{31} &a_{32} &a_{33} &a_{34} \end{bmatrix} = \begin{bmatrix} 2 &3 &1 &-1 \5 &1 &1 &9 \ 3 &2 &4 &11 \end{bmatrix}$ The new iteration co-efficient matrix is calculated as $A'=\begin{bmatrix} a'_{11}=a_{11} &a'_{12}=\frac {a_{12}}{a_{11}'} &a'_{13}= \frac {a_{13}}{a'_{11}} &a'_{14}=\frac {a_14}{a'_{11}} \ a'_{21}=a_{21} &a'_{22}=a_{22}-a'_{21}a'_{12} & a'_{23}= \frac {a_23-a'_{21}a'_{13}}{a'_{22}} &a'_{24}= \frac {a_{24}-a'_{21}a'_{14}}{a'_{22}} \ a'_{31}=a_{31}&a'_{32}=a_{32}-a'_{31}a'_{12} &a'_{33}= a_{33}- a'_{31}a'_{13}- a'_{32}a'_{23} &a'_{34} = \frac {a_{34}-a'_{31}a'_{14}-a'_{32}a'_{24}}{a'_{33}} \end{bmatrix}$ $A'=\begin{bmatrix} 2 &\frac {3}{2} & \frac {1}{2} & -\frac {1}{2} \ 5&1-5\times \frac {3}{2}= -\frac {13}{2} & -\frac {2 \left ( 1-5 \left ( \frac {1}{2} \right ) \right )}{13} = \frac {3}{13} & -\frac {2 \left ( 9+ \frac {5}{2} \right )}{13}= -\frac {23}{13} \ 3&2-3\times \frac {3}{2} = -\frac {5}{2}&4-\frac {3}{2}- \left ( - \frac {5}{2} \right ) \left ( \frac {3}{13} \right )\approx 3.077 &\frac {11-3 \left ( -\frac {1}{2} \right )- \left ( - \frac {5}{2} \right )\left ( - \frac {23}{13} \right )}{3.077} =2.625 \end{bmatrix}$ $z=a'_{34}\approx 2.625 \ y=a'_{24}- a'_{23}z = - \dfrac {23}{13}- \dfrac {3}{13} (2.625) \approx -2.375 \ x=a'_{14} - a'_{13}z - a'_{12}y = - \dfrac {1}{2} - \dfrac {1}{2} (2.625) - \dfrac {3}{2} (-2.375)=1.75$

Hence, by Crout's Method, (x,y,z)= (2.625, -2.375, 1.75)