...or prove that none exists.

Note that such a matrix $M$ couldn't be primitive, so there would be at least one entry equal to zero in every power $M^k$ (Perron-Frobenius theory).

Preferably the matrix would have a diagonal that is not all zero, and at the risk of making the problem imprecise, I'd like to find such a matrix with as few zeros and ones as possible.

Thank you.