$\begin{bmatrix} 1 & 2 &0 \\ 3 & c &1 \\ 0 &1 &1 \end{bmatrix}$
$=\begin{bmatrix} 1 & 0 & 0\\ l_{21}& 1 &0 \\ l_{31}& l_{32} & 1 \end{bmatrix}$.$\begin{bmatrix} u_{11} & u_{12} & u_{13}\\ 0& u_{22} &u_{23} \\ 0& 0 & u_{33} \end{bmatrix}$
$u_{11}=1$
$u_{12}=2$
$u_{13}=0$
$u_{21}=3$
$u_{22}=c-6$
$u_{23}=1$
$l_{31}=0$
$l_{32}=\frac{1}{c-6}$
$u_{33}=1-\frac{1}{c-6}$
So, Now lower triangular matrix is
$=\begin{bmatrix} 1 & 0 & 0\\ 3& 1 &0 \\ 0& \frac{1}{c-6}& 1 \end{bmatrix}$
And upper triangular matrix
$\begin{bmatrix} 1 & 2 & 0\\ 0& c-6 &1 \\ 0& 0 & 1-\frac{1}{c-6} \end{bmatrix}$
So, denominator cannot be $0$, So, c cannot be $6$