# Let M be any $3\times3$ matrix with entries from the set { 0,1, 2}. The maximum number of such matrices, for which the sum of diagonal elements of MTM is seven, is_________ Option: 1 540 Option: 2 363 Option: 3 450 Option: 4 336

$\text{Let }M=\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right]$

$M^{T}=\left[\begin{array}{lll} a & d & g \\ b & e & h \\ c & f & i \end{array}\right]$

$M^TM=\left[\begin{array}{lll} a & d & g \\ b & e & h \\ c & f & i \end{array}\right]\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right]$

Sum of diagonal matrix MTM

$\mathrm{a}^{2}+\mathrm{b}^{2}+\mathrm{c}^{2}+\mathrm{d}^{2}+\mathrm{e}^{2}+\mathrm{f}^{2}+\mathrm{g}^{2}+\mathrm{h}^{2}+\mathrm{i}^{2}=7$

$\mathbf{Case\; I : Seven\; (1's)\; and\; two \;(0's)}$

$\frac{9 !}{7 ! 2 !}=36$

$\mathbf{Case\; II : One\;(2)\;Three\; (1's)\; and\; Five \;(0's)}$

$\frac{9 !}{5 ! 3 !1!}=504$

Total = 540

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