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

$\begin{bmatrix}a&b&c\\ d&e&f\\ g&h&i\end{bmatrix}\begin{bmatrix}a&d&g\\ b&e&h\\ c&f&i\end{bmatrix}$
$a^{2}+b^{2}+c^{2}+d^{2}+e^{2}+f^{2}+g^{2}+h^{2}+i^{2}=7$
Case-I : Seven (1's) and two (0's)
${ }^{9} C _{2}=36$
Case-II : One (2) and three (1's) and five (0's)
$\frac{9 !}{5 ! 3 !}=504$
$\therefore $ Total $=540$