Let S be the set containing all 3 × 3 matrices with entries from - 1 , 0 , 1 . The total number of matrices A ∈ S such that the sum of all the diagonal elements of A T A is 6 is ______.

Step-by-step solution

Given that the sum of all the diagonal elements of A T A is 6 , means sum of all diagonal elements is equal to sum of square each element of the matrix. i.e., A = a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 and A T = a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 Then t r A · A T = a 1 2 + a 2 2 + a 3 2 + b 1 2 + b 2 2 + b 3 2 + c 1 2 + c 2 2 + c 3 2 Given a i , b i , c i ∈ - 1 , 0 , 1 for i = 1 , 2 , 3 Now sum to be 6 exactly three of them should be zero and rest should be 1 or - 1 . So, total number of possible matrices