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 $\mathrm{M}^{\top} \mathrm{M}$ is seven, is
Solution:
$\left[\begin{array}{lll}a & b & c \\ d & e & f \\ g & h & i\end{array}\right]\left[\begin{array}{lll}a & d & g \\ b & e & h \\ c & f & i\end{array}\right]$
$a^{2}+b^{2}+c^{2}+d^{2}+e^{2}+f^{2}+g^{2}+h^{2}+i^{2}=7$
Case I : Seven (l's) and two (O's)
${ }^{9} \mathrm{C}_{2}=36$
Case II : One (2) and three (1's) and five (O's)
$\frac{9 !}{5 ! 3 !}=504$
$\therefore$ Total $=540$
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