The Universality Problem is the dual of the emptiness problem: is L(A) = Σ∗?
It can be solved by minor variations of any one of the algorithms for Emptiness or (with a little less work) it can simply be reduced to Emptiness.
Theorem (Universality) The Universality Problem for Regular Languages is decidable.
Proof: L(A) = Σ*⇔ L(A) = ∅. As regular languages are effectively closed under complement we can simply build the DFA for the complement of L(A) and ask if it recognizes the empty language.