How many 3x3 matrices M with entries from (0,1,2) are there, for which the sum of the diagonal entries of M.T* M is 5?

Dear Student,
Please find below the solution to the asked query:

LetM=abcpqrxyzM'=apxbqycrzNow we are just interested in diagonal elements of MM' hence we willjust write diagonal elements.MM'=abcpqrxyz apxbqycrzMM'=a2+b2+c2p2+q2+r2x2+y2+z2Sum of diaginal entries=a2+b2+c2+p2+q2+r2+x2+y2+z25 can be achived by giving 1 to 5 elements and 0 to 4 elementsNumber of ways=9C5orgives 2 to one element and 1 to other and 0 to remaining 7 22+12=5Number of ways=2! ×9C2Total ways=9C5+2! ×9C2

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