MathDB

For each positive integer

n

, let

M(n)

be the

n\times n

matrix whose

(i,j)

entry is equal to

1

i+1

is divisible by

j

, and equal to

0

otherwise. Prove that

M(n)

is invertible if and only if

n+1

is square-free. (An integer is square-free if it is not divisible by a square of an integer larger than

1

Problem Statement