Let M be any 3 × 3 matrix with entries from the set {0, 1, 2}. Then maximum number of such matrices, for which the sum of diagonal elements of $M^{T}M$ is seven, is ______ 

$\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]\left[\begin{array}{ccc}a& d& g\\ b& e& h\\ c& f& i\end{array}\right]=a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2+i^2=7$

Case I : Seven (1's) and two (0's) ${ }^9{C}_2=36$

and Case- II : One (2) and three (1’s) and five (0’s) - $\frac{9!}{5!3!}=504$

Therefore, total is $36+504=540$