Raising rent from zero (and an apology)

[Christopher Olsen]

Yesterday I posted this question, but it was never answered because my original post was long and wordy. So here is the less wordy version of my question. My apologies. 

Let's say you acquire a house where there is a tenant at will who has no rental agreement and is paying zero rent. The market rate for this property is $2000-2500 a month. You are also covering all utilities.

This happened because the previous owner/resident let her live there for free and was quite right in the head and then died, and you were either dumb enough to purchase the property or unlucky enough to inherit it (in this case, the latter).

The tenant gets $900 a month social security, but is probably not too old to work.

Can the new owner raise her rent from zero to $2000 a month? Obviously, she cannot pay that amount. But will she still owe that money after she is, at some point, legally evicted? 

For argument's sake, let's assume that this person is an otherwise respectable individual who probably does not want her credit to get any worse than it already is from being evicted and further in debt.

Thank you.