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Define-a-3-3-matrix-whose-entries-are-the-first-9-positive-integers-Let-s-k-be-the-sum-of-the-elements-across-the-kth-row-Is-there-such-a-matrix-where-s-1-s-2-s-3-1-2-3-




Question Number 8236 by Yozzias last updated on 03/Oct/16
Define a 3×3 matrix whose entries  are the first 9 positive integers.  Let s_k  be the sum of the elements  across the kth row. Is there such a   matrix where s_1  : s_2  : s_3  = 1 : 2 : 3 ?  −−−−−−−−−−−−−−−−−−−−  What about n×n matrices whose  elements are the first n^2  positive  integers? Is there a matrix such  that s_1  : s_2  : s_3  : s_4  :.....: s_n = 1 : 2 : 3 :...: n?
Definea3×3matrixwhoseentriesarethefirst9positiveintegers.Letskbethesumoftheelementsacrossthekthrow.Istheresuchamatrixwheres1:s2:s3=1:2:3?Whataboutn×nmatriceswhoseelementsarethefirstn2positiveintegers?Isthereamatrixsuchthats1:s2:s3:s4:..:sn=1:2:3::n?
Commented by Rasheed Soomro last updated on 03/Oct/16
s_1 +s_2 +s_3 =((9×(9+1))/2)=45    [∵ s_1 ∪ s_2 ∪s_3 ={1,2,3,...,9}]  If 45 is divided in 1:2:3  s_1 =((45)/6) =((15)/2) .But sum of three whole numbers may  not be fraction.  So such 3×3 matrix  is not possible.    Similarily for n×n matrix  s_1 +s_2 +s_3 +...+s_n =((n^2 (n^2 +1))/2) [∵  s_1 ∪ s_2 ∪s_3 ...∪s_n ={1,2,3,...,n^2 }  If  s_1  : s_2  : s_3  : s_4  :.....: s_n = 1 : 2 : 3 :...: n  s_1 =(((n^2 (n^2 +1))/2)/((n(n+1))/2)) must be whole number.  s_1 =((n^2 (n^2 +1))/2)×(2/(n(n+1)))=((n(n^2 +1))/(n+1))  But this is not whole number in geneal.  So in general such n×n matrix is not possible.
s1+s2+s3=9×(9+1)2=45[s1s2s3={1,2,3,,9}]If45isdividedin1:2:3s1=456=152.Butsumofthreewholenumbersmaynotbefraction.Sosuch3×3matrixisnotpossible.Similarilyforn×nmatrixs1+s2+s3++sn=n2(n2+1)2[s1s2s3sn={1,2,3,,n2}Ifs1:s2:s3:s4:..:sn=1:2:3::ns1=n2(n2+1)2n(n+1)2mustbewholenumber.s1=n2(n2+1)2×2n(n+1)=n(n2+1)n+1Butthisisnotwholenumberingeneal.Soingeneralsuchn×nmatrixisnotpossible.
Commented by Yozzias last updated on 03/Oct/16
Thanks!
Thanks!
Answered by Yozzias last updated on 04/Oct/16
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