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


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Question Number 8236 by Yozzias last updated on 03/Oct/16

$Define a 3 \times 3 matrix whose entries$ $are the first 9 positive integers .$ $Let s_{k} be the sum of the elements$ $across the kth row . Is there such a$ $matrix where s_{1} : s_{2} : s_{3} = 1 : 2 : 3 ?$ $- - - - - - - - - - - - - - - - - - - -$ $What about n \times n matrices whose$ $elements are the first n^{2} positive$ $integers ? Is there a matrix such$ $that s_{1} : s_{2} : s_{3} : s_{4} : . . . . . : s_{n} = 1 : 2 : 3 : . . . : n ?$
Commented by Rasheed Soomro last updated on 03/Oct/16
$s_1 +s_2 +s_3 =((9×(9+1))/2)=45 [∵ s_1 ∪ s_2 ∪s_3 ={1,2,3,...,9}] If 45 is divided in 1:2:3 s_1 =((45)/6) =((15)/2) .But sum of three whole numbers may not be fraction. So such 3×3 matrix is not possible. Similarily for n×n matrix s_1 +s_2 +s_3 +...+s_n =((n^2 (n^2 +1))/2) [∵ s_1 ∪ s_2 ∪s_3 ...∪s_n ={1,2,3,...,n^2 } If s_1 : s_2 : s_3 : s_4 :.....: s_n = 1 : 2 : 3 :...: n s_1 =(((n^2 (n^2 +1))/2)/((n(n+1))/2)) must be whole number. s_1 =((n^2 (n^2 +1))/2)×(2/(n(n+1)))=((n(n^2 +1))/(n+1)) But this is not whole number in geneal. So in general such n×n matrix is not possible.$
$s_{1} + s_{2} + s_{3} = \frac{9 \times (9 + 1)}{2} = 45 [∵ s_{1} \cup s_{2} \cup s_{3} = {1, 2, 3, . . ., 9}]$ $If 45 is divided in 1 : 2 : 3$ $s_{1} = \frac{45}{6} = \frac{15}{2} . But sum of three whole numbers may$ $not be fraction .$ $So such 3 \times 3 matrix is not possible .$ $Similarily for n \times n matrix$ $s_{1} + s_{2} + s_{3} + . . . + s_{n} = \frac{n^{2} (n^{2} + 1)}{2} [∵ s_{1} \cup s_{2} \cup s_{3} . . . \cup s_{n} = {1, 2, 3, . . ., n^{2}}$ $If s_{1} : s_{2} : s_{3} : s_{4} : . . . . . : s_{n} = 1 : 2 : 3 : . . . : n$ $s_{1} = \frac{\frac{n^{2} (n^{2} + 1)}{2}}{\frac{n (n + 1)}{2}} must be whole number .$ $s_{1} = \frac{n^{2} (n^{2} + 1)}{2} \times \frac{2}{n (n + 1)} = \frac{n (n^{2} + 1)}{n + 1}$ $But this is not whole number in geneal .$ $So in general such n \times n matrix is not possible .$
Commented by Yozzias last updated on 03/Oct/16

$Thanks!$
	Answered by Yozzias last updated on 04/Oct/16

	$Answer in comments .$
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