The total number of 3 × 3 matrices A having enteries from the set 0 , 1 , 2 , 3 such that the sum of all the diagonal entries of A A T is 9 , is equal to

Step-by-step solution

Let A = a b c d e f g h i ,   A T = a d g b e h c f i Diagonal elements of A A T   is   a 2 + b 2 + c 2 ,   d 2 + e 2 + f 2 ,   g 2 + b 2 + c 2 .   Sum = a 2 + b 2 + c 2 + d 2 + e 2 + f 2 + g 2 + h 2 + i 2 = 9 a , b , c , d , e , f , g , h , i ∈ 0 , 1 , 2 , 3 Case No. of Matrices (1) All – 1 ' s 9 ! 9 ! = 1 (2) One → 3 ' s , remaining- 0 ' s 9 ! 1 ! × 8 ! = 9 (3) One- 2 ' s Five- 1 ' s Three- 0 ' s 9 ! 1 ! × 5 ! × 3 ! = 8 × 63 (4) Two - 2 ' s O