I'm trying to determine if a certain class of matrices are M-matrices in general. I'm considering square matrices $A$ with the following properties:
- $\det(A) > 0$ (strictly),
- all the diagonal entries are positive,
- all the off diagonal entries are negative.
An M-matrix can be characterized in many ways. I've tried proving this (or finding a counterexample) by looking at the principal minors and have found that $A$ is an M-matrix if it has dimension 2 or 3, but it's hard to make any sort of induction with that. Right now I'm trying two other definitions (they're equivalent) of M-matrices
- There is a positive vector such that $Ax > 0$ (component-wise).
- $A$ is monotone (i.e. $Ax \geq 0$ implies $x \geq 0$).
Again, 1 isn't hard to show if the matrix is small, but this is hard to generalize, so I thought an easier approach might be using 2 and try to proceed by contradiction. Does anyone here have some suggestions? This is an outside project for a class I'm working on so I don't know if these matrices are or are not M-matrices in general - mostly just looking for tips here.