# Gregory, Steven A

## inverse of an integer matrix ★★

Author(s): Gregory

\begin{question} I've been working on this for a long time and I'm getting nowhere. Could you help me or at least tell me where to look for help. Suppose D is an m-by-m diagonal matrix with integer elements all $\ge 2$. Suppose X is an m-by-n integer matrix $(m \le n)$. Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.

\end{question}

Keywords: invertable matrices, integer matrices