MathematicsLet 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 MTM is seven, is

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

Solution:

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

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