For a positive integer n>1, let f(n) be the largest real number such that for every n×n diagonal matrix M with positive diagonal entries, if tr(M)<f(n), then M-J is invertible. Determine f(n). (The matrix J is the square matrix with all entries 1.)

The best solution was submitted by Kyoungseok Jang(장경석), 2011학번. Congratulations!

Here is his Solution of Problem 2011-14.

Alternative solutions were submitted by 곽영진 (2011학번, +3), 박민재 (2011학번, +3), 라준현 (수리과학과 2008학번, +3), 서기원 (수리과학과 2009학번, +3), 배다슬 (수리과학과 2008학번, +3), 김범수 (수리과학과 2010학번, +3), 어수강 (서울대학교 수리과학부 대학원, +3), 조위지 (Stanford Univ. 물리학과 박사과정, +3).

PS. There were solutions without computing the determinant. Here is a Solution of Problem 2011-14 by 김범수.