Let M be any 3 x 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^TM is seven, is

${\mathrm{M}}^{\mathrm{T}}\mathrm{M}=\left[\begin{array}{l}\mathrm{a}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{b}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{c}\\ \mathrm{d}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{e}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{f}\\ \mathrm{g}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{h}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{i}\end{array}\right]\left[\begin{array}{l}\mathrm{a}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{d}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{g}\\ \mathrm{b}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{e}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{h}\\ \mathrm{c}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{f}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{i}\end{array}\right]={\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2+{\mathrm{d}}^2+{\mathrm{e}}^2+{\mathrm{f}}^2+{\mathrm{g}}^2+{\mathrm{h}}^2+{\mathrm{i}}^2=7$
Case I: Seven (1’s) and two (0’s)Number of matrices$=\frac{9!}{7!2!}=36$
Case II : One (1) and three (2’s) and five (0’s)Number of matrices $=\frac{9!}{5!3!}=504$
$\therefore$Total number of matrices = 540