Finding whether or not a counterexample exists Does a counterexample exist for the following argument?

If person A is not home, then person B is. But, if A is not home, then B isn’t. So, they are both home.

Translated to logical notation:
1) $\neg A \to B$
2) $\neg A \to \neg B$
3) $\therefore A \land B$
To my understanding, a counterexample is when the premises are true but the conclusion is false. I've equated them as such, but I'm stuck in proving whether or not there is a contradiction, since there are too many cases to deal with (e.g. $A \land B \equiv F$ has 3 cases). How would I find out whether there's a counterexample or not?
1) and 2) seem contradicting already, but they also have 3 cases each.
 A: We can say, from (1) and (2), that:
$$\lnot A\to (B\land \lnot B)$$
As $B\land\lnot B=F$, we have:
$$\lnot A\to F$$
A: 
To my understanding, a counterexample is when the premises are true but the conclusion is false.

Correct; more correctly, a counterexample invalidates an argument by making all premises true while falsifying the conclusion.
Here the conclusion is false if either of $A$ or $B$ are false.
Both premises are implications, which are held to be false only if the antecedent is true while the consequent is false.
Since the consequent of both implications are contradictory, then both implications can only be true at the same time if their antecedent is false.   Hence if a counterexample is to be found, $A$ must be true.  
The way to make the conclusion ($A\wedge B$) false when $A$ is true is to make $B$ false.   And lo and behold: when $A$ is true and $B$ false then both premises are true.   Thus we have found a counterexample.
Therefore the argument $\neg A\to B, \neg A\to\neg B\vdash A\wedge B$ is proven to be invalid, by way of counterexample.
