For each positive integer n, let Mn be the square matrix (nxn) where each diagonal entry is 2018, and every other entry is 1.
Determine the smallest positive integer n (if any) for which the value
of det(Mn) is a perfect square.
I ran through the det(M2) to det(M7), this time using the correct formulae for the determinants. :-)
det(M7) = x^7 -21 x^5 +70 x^4 -105 x^3 +84 x^2 -35 x + 6
(Thanks Dr. Wolfram)
I didn't get 2018 to work for n<8 but I think maybe 2019 works for M7
I can not be sure that there is not a roundoff error (could someone with this computer ability check?) but perhaps for 2019
(Or else I was just really unlucky to get 5 zeros.)
But, I am interested to know how to solve this problem more elegantly!
Edited on June 15, 2018, 7:17 pm
Edited on June 15, 2018, 7:18 pm