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Question

How many 3×3 matrices M with entries from {0,1,2} are there, for which the sum of the diagonal entries of MTM is 5?
  1. 126
  2. 198  
  3. 162 
  4. 135 


Solution

The correct option is B 198  
MTM=abcdefghiadgbehcfi

Sum of diagonal entries of MTM
=a2+b2+c2+d2+e2+f2+g2+h2+i2=5

Two possible case:
(1) Five "1's" and four "0's" 
Number of ways = 9C5=126

(2) One "1" and one "2" and seven "0's" 
Number of ways= 9C7× 2C1× 1C1=72
Total number of possible matrices =126+72=198

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