The number of all 3 × 3 matrices A , with entries from the set - 1 , 0 , 1 such that the sum of the diagonal elements of A A T is 3 , is ___________.

Step-by-step solution

Let, A = a i i 3 × 3 t r ( A A T ) = 3 a 11 2 + a 12 2 + a 13 2 + a 21 2 + . . . . . . . + a 33 2 = 3 Possible cases 0 , 0,0 , 0,0 , 0,1 , 1,1 → 1 0 , 0,0 , 0,0 , 0 , - 1 , - 1 , - 1 → 1 0 , 0,0 , 0,0 , 0,1 , 1 , - 1 → 3 0 , 0,0 , 0,0 , 0 , - 1,1 , - 1 → 3   9 C 6 × 8 = 84 × 8 = 672