The prime minister is the member of parliament that commands the confidence of the House of Commons.
At least that is one of the informal definitions. It also seems to be tied to leadership of the largest party, but that seems to be a function of the confidence requirement, since it is difficult to maintain confidence of the House when the largest party is withholding it.
Difficult but not impossible.
So if the Liberals and the NDP cobble together a coalition, it will be, by definition, unstable, with the Conservatives never offering a vote of confidence in favour of that coalition. But the Bloc Quebecois could keep that coalition alive.
But only if the Bloc Quebecois votes to support that coalition.
Combined, the Liberals and the NDP have 113 seats (subtracting Speaker Peter Milliken's vote). That is less than the 146 votes held by the Conservatives and the two independents (who would likely vote with the Conservatives). So the Bloc Quebecois, which by all accounts will not formally join the coalition, will keep that coalition in power on a vote by vote basis if the Bloc Quebecois is willing to lend the coalition 33 of its 49 votes.
That is a lot of power in the hands of the Bloc Quebecois, and that is a huge chunk of power to hand over to a party that is (1) concentrated by design in exactly one province and (2) is dedicated to showing how Canada's government is dysfunctional and so promote Quebec separatism.
How is this coalition going to claim to be able to form a stable government, held hostage by the separatist party and representing a mere 21 seats of the 92 seats west of Ontario? The West is a region that is notoriously short of patience when it comes to pandering to Quebec parochial interests.
The reserve power in the hands of the Governor General is broad and ill-defined. Refusing to dissolve parliament in the wake of a non-confidence vote is one of them, and given that it has barely been two months since the last election, the Governor General will be under a lot of pressure to avoid another election.
But the Conservatives came into power with a stronger majority, not a weaker one. The Liberals dropped to a historic low in support. To put the Liberals into power is likely legal, but it flies in the face of common sense. And the coalition itself is insufficient in size to command the confidence of parliament without the support of a party that is not a member of that coalition, a party that is a force for instability in Canada, and designed to be so.
If this was a numbers game, we wouldn't need a Governor General. A simple computer program could be used to assign power. But it isn't a numbers game. The Governor General is being called upon to use her powers to give Canadians a stable government that represents their interests. She must ask herself whether this Liberal-NDP coalition propped up by the Bloc Quebecois, a coalition that does not include the largest party in the House, truly represents the will of Canadians. Is it likely to serve Canadians well, or is it likely to exacerbate regional frustrations? And is it likely to serve Canadians for any reasonable length of time, or will it crumble within weeks?
Remember, on paper, there is a party that represents 142 seats versus a coalition that represents 123 seats. The Bloc Quebecois and those two independents are not part of the coalition. In effect, they are refusing to commit to voting one way or the other. If the Liberal-NDP coalition added up to more seats than the Conservatives, than the situation would be different. But this coalition would represent a weaker government, not a stronger one. We'd be moving away from stability.
If this coalition is likely to fail, then maybe it makes sense just to go to the polls again.