Hi!
Could anyone please help me with these questions?
1.) Prove that the matrix 0 0 0
0 0 0
1 0 0
does not have a cube root.
(Hint: What would the eigenvalues of a hypothetical cube root be?)
2.) One of the following sets is countable, and the other one is not:
a) The set of all permutations p: N --> N such that p(n)=n for all but finitely many n
b) The set of all permutations p: N --> N such that /p(n) - n/ <= 1 for all n.
(with <= I mean smaller than or equal and the lines are mod signs)
I would think that the first set is countable and the second set isn't but I can't really give a formal prove for that one.
Thanx a lot for your help.
Andi