Question

How many 3 x 3 matrices M with entries from {0, 1, 2} are there, for which the sum of the diagonal entries of M^TM is 5 ? 

(A) 126 

(B) 198 

(C) 162 

(D) 135

Answer 1

Correct option is (B) 198 

Let matrix M = [k_ij]

Then sum of diagonal entries = \(\displaystyle\sum_{}^{} k^2_{ij}\)

⇒ \(\displaystyle\sum_{}^{} k^2_{ij} = 5\)

where k_ij are from {0, 1, 2}

⇒ Total number of matrices = ⁹C₅ + ⁹C₁ . ⁸C₁ = 198