Consider the matrix $A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$  where  $a_{ij}\in \{1,2,3,4,5,6\}$  and cofactors  $a_{ij}=0,\forall i,j.$If no two elements of same row are same , then the total number possible matrices of  A is/are ____

$\left|\begin{array}{cc}a& b\\ d& e\end{array}\right|=0\Rightarrow \frac{a}{d}=\frac{b}{e}$  and $\left|\begin{array}{cc}b& c\\ e& f\end{array}\right|=0\Rightarrow \frac{b}{e}=\frac{c}{f}$
 $\Rightarrow \frac{a}{d}=\frac{b}{e}=\frac{c}{f}$
similarly  $\frac{d}{g}=\frac{e}{h}=\frac{f}{i}and\frac{a}{g}=\frac{b}{h}=\frac{c}{i}$
a:b:c=d:e:f=g:h:i
Case-I $\{\text{a},\text{b},\text{c}\}=\{\text{d},\text{e},\text{f}\}=\{\text{g},\text{h},\text{i}\}$  then total matrices possible equals to ${ }^6{\text{C}}_3\times 3!=120$ 
Case-II two rows the same and one different e.g. $\{\text{a},\text{b},\text{c}\}=\{1,2,3\}$  and    $\{\text{d},\text{e},\text{f}\}=\{\text{g},\text{h},\text{i}\}$ $=\{2,4,6\}$
the total matrices possible equals to ${ }^3{\text{C}}_2\times 3!\times 2$ =36, total 156