TITLE: Matrices with elements that are a distinct set of prime numbers: always invertible?
QUESTION [5 upvotes]: Inspired by a previous question, given a square non-symmetric matrix whose elements are all prime but distinct from each other, does this guarantee that the matrix is invertible? It's easy to see $N=2$ this holds, a counter-example would imply that there must be four distinct primes such that $p_1 p_2 = p_3 p_4$.

REPLY [3 votes]: A $4 \times 4$ counterexample is
$$\pmatrix{43 & 17 & 53 & 19 \cr
11 & 71 & 3 & 89 \cr
79 & 47 & 29 & 37 \cr
31 & 7 & 61 & 13 \cr
}$$