Why $ always matches end of line

The following is a simple piece of code that demonstrates the apparently erroneous behavior of line-ending matching ("$") in .Net regular expressions. Am I missing something?

        string input = "Hello\nWorld\n";
        string regex = @"^Hello\n^World\n";  //Match
        //regex = @"^Hello\nWorld\n";  //Match
        //regex = @"^Hello$";  //Match
        //regex = @"^Hello$World$";  //No match!!!
        //regex = @"^Hello$^World$";  //No match!!!

        Match m = Regex.Match(input, regex, RegexOptions.Multiline | RegexOptions.CultureInvariant);
        Console.WriteLine(m.Success);