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How many 3xx3 matrices M with entries fr...

How many `3xx3` matrices M with entries from `{0, 1, 2}` are there, for which the sum of the diagonal entries of `M^(T)M` is 5 ?

A

198

B

162

C

126

D

135

Text Solution

Verified by Experts

The correct Answer is:
A
Sum of diagonal entries of `M^(T) M` is `sum a_(i)^(2).`
`overset(9)underset(i - 1)sum a_(i)^(2) = 5`
Possibilities
`I. 2, 1, 0, 0, 0, 0, 0, 0, 0` which gives `(9!)/(7!)` metrices
`II. 1, 1, 1, 1, 1, 0, 0, 0, 0,` which gives `(9!)/(4! xx 5!)` metrices
Total matrices `= 9 xx 8 + 9 xx 7 xx 2 = 198`
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