The number of 3 × 3 matrices M with entries from 0,1 , 2 , such that the sum of the diagonal elements of M T M is 5 , are

Step-by-step solution

Let, M = a b c d e f g h i ∴ Sum of the diagonal elements, Tr ( M T ​   M ) = a 2 + b 2 + c 2 + d 2 + e 2 + f 2 + g 2 + h 2 + i 2 = 5 , where entries are 0 , 1 , 2 Only two cases are possible. (I) Five entries are 1 and other fourare 0 ∴         9 C 5 × 1 (II) One entry is 2 , one entry is 1 and others are 0 . ∴         9 C 2 × 2 Total cases = 126 + 72 = 198 .