The total number of   $3\times 3$  matrices A having entries from the set  $\{0,1,2\}$  such that the sum of all the diagonal entries of  $A^{T}A$  is 5 is _____

 Let  $A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right],a,b,c,d,e,f,g,h,i\in \{0,1,2,3,4\}$  sum of diagonal elements in  $A^{T}A$  is 5 $\Rightarrow a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2+i^2=5$
The elements of A may be
(i) one 2, one 1, seven 0’s
(ii) five 1’s, four 0’s
$\therefore$  The total no of matrices formed is  
 $\frac{9!}{7!}+\frac{9!}{5!4!}=198$