Is the norm of a singular matrix necessarily zero?

Question

If not, can you give example of when it's not?

Answer 1

Take a diagonal matrix with one zero at the diagonal and the rest of the entries in the diagonal being 1. This matrix is singular, but has positive norm.

Answer 2

Well a matrix has zero norm if and only if it is the zero matrix so, can you think of any matrix which isn't all zeros, but which also isn't invertible?