/sci/ - Science & Math

Can there be a real parameter [math]p[/math] and (not necessarily square) matrix [math]M[/math] such that all of the following are satisfied?
> All matrix entries are bounded by [math]0\leq M_{ij}\leq p[/math]
> For every row [math]i[/math], [math]\sum_{j}M_{ij}=1[/math] (so [math]M[/math] is a probability matrix)
> For every column [math]j[/math], [math]M_{ij}=0[/math] for at least [math]p\times 100\%[/math] of the [math]i[/math]'s (a kind of sparsity condition)
I suspect that the answer is no, but I haven't been able to come up with a proof or a counterexample.