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?126198  162 135

Solution

The correct option is B 198  MTM=⎡⎢⎣abcdefghi⎤⎥⎦⎡⎢⎣adgbehcfi⎤⎥⎦ 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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