The number of 3 × 3 matrices M with entries from 0,1,2 for which the sum of the diagonal entries of MTM is 5, are

Question

The number of 3 × 3 matrices M with entries from 0,1,2 for which the sum of the diagonal entries of MTM is 5, are (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] Sum of diagonal elements, a2 + b2 + c2 + d2 + e2 +f2 + g2 + h2 + i2 = 5 Case - I Five (1's) and four (0's) 9C5 = 126 Case - II: One (2) and one (1) 9C2 = 2! = 72 ∴ Total =198

Q. The number of $3 \times 3$ matrices $M$ with entries from ${0, 1, 2}$ for which the sum of the diagonal entries of $M^{T} M$ is $5$ , are

$⎣ ⎡ a d g b e h c f i ⎦ ⎤ ⎣ ⎡ a b c d e f g h i ⎦ ⎤$
Sum of diagonal elements,
$a^{2} + b^{2} + c^{2} + d^{2} + e^{2} + f^{2} + g^{2} + h^{2} + i^{2} = 5$
Case - I Five (1's) and four (0's)
$^{9} C_{5} = 126$
Case - II : One (2) and one (1)
$^{9} C_{2} = 2! = 72$
$∴$ Total $= 198$

Q. The number of 3×3 matrices M with entries from {0,1,2} for which the sum of the diagonal entries of MTM is 5, are

⎣⎡​adg​beh​cfi​⎦⎤​⎣⎡​abc​def​ghi​⎦⎤​ Sum of diagonal elements, a2+b2+c2+d2+e2+f2+g2+h2+i2=5 Case - I Five (1's) and four (0's) 9C5​=126 Case - II : One (2) and one (1) 9C2​=2!=72 ∴ Total =198

Q. The number of $3 \times 3$ matrices $M$ with entries from ${0, 1, 2}$ for which the sum of the diagonal entries of $M^{T} M$ is $5$ , are