2011-14 Invertible matrices

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.)

(Due to a mistake, the problem is fixed at 3:30PM Friday.)

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5 thoughts on “2011-14 Invertible matrices

  1. Sukang

    diagonal matrix M with positive diagonal entries 라는 말이
    M은 모든 대각성분이 양수인 대각행렬이다 맞나요?^^;;

  2. Sukang

    문제가 잘못됐던 거군요. I로 놓으니까 문제가 1분만에 풀리더라구요. ㅎㅎ
    I면 f(n)=1이 되는거 맞나요?ㅎㅎ

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