Prime minister is the leader of the political party with a majority of seats in the Senate?

No, in both Ireland and Canada, the Prime Minister leads the largest party in the lower house (Dail in Ireland and House of Commons in Canada). In Australia the prime minister is the leader of the political group with a mayority in the lower house / house of representatives. The leader of the senate is caled the president of the senate. the president of the senate doesnot necesarily have to be of the same party as that of the prime minister.