Let M be any 3 × 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 .

Question

Let M be any 3 × 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 . (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

[a&b&c d&e&f g&h&i][a&d&g b&e&h c&f&i] a2+b2+c2+d2+e2+f2+g2+h2+i2=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) (9 !/5 ! 3 !)=504 ∴ Total =540

Q. 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 _______.

$⎣ ⎡ a d g b e h c f i ⎦ ⎤ ⎣ ⎡ a b c d e f g h i ⎦ ⎤$
$a^{2} + b^{2} + c^{2} + d^{2} + e^{2} + f^{2} + g^{2} + h^{2} + i^{2} = 7$
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)
$\frac{9 !}{5 ! 3 !} = 504$
$∴$ Total $= 540$

Q. Let M be any 3×3 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 _______.

⎣⎡​adg​beh​cfi​⎦⎤​⎣⎡​abc​def​ghi​⎦⎤​ a2+b2+c2+d2+e2+f2+g2+h2+i2=7 Case-I : Seven (1's) and two (0's) 9C2​=36 Case-II : One (2) and three (1's) and five (0's) 5!3!9!​=504 ∴ Total =540

Q. 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 _______.

$⎣ ⎡ a d g b e h c f i ⎦ ⎤ ⎣ ⎡ a b c d e f g h i ⎦ ⎤$
$a^{2} + b^{2} + c^{2} + d^{2} + e^{2} + f^{2} + g^{2} + h^{2} + i^{2} = 7$
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)
$\frac{9 !}{5 ! 3 !} = 504$
$∴$ Total $= 540$