Question

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

Answer 1

Let $M=\left[\begin{array}{ccc}a& d& g\\ b& e& h\\ c& f& i\end{array}\right]$
$\therefore M^{T}M=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]\left[\begin{array}{ccc}a& d& g\\ b& e& h\\ c& f& i\end{array}\right]$
$\Rightarrow a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2+i^2=7$
Case $1:$ Seven $\left(1^{\prime }s\right)$ and two $\left(0^{\prime }s\right)$
${}^9{C}_2=36$
Case $2:$ One $\left(2\right)$ and three $\left(1^{\prime }s\right)$ and five $(0,s)$
$\frac{9!}{1!3!2!}=504$

$\therefore$ Total $=540$