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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?

A
126
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B
198
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C
162
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D
135
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Solution

## The correct option is A 198Let M be a 3×3 square matrix,M=⎡⎢⎣abcdefghi⎤⎥⎦Thus, MTM=⎡⎢⎣adgbehcfi⎤⎥⎦⎡⎢⎣abcdefghi⎤⎥⎦The sum of the diagonal elements of MTM will be S=a2+b2+c2+d2+e2+f2+g2+h2+i2For S=5⇒a2+b2+c2+d2+e2+f2+g2+h2+i2=5 out of {0,1,2}, there are only two possibilities:1. One element is 2, one element is 1 and the rest are 0 =9C22!=722. 5 elements are 1 and the rest are all 0 =9C5=126Hence, total number of matrices M =72+126=198Hence, option B is correct.

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