Let A be any n × n complex matrix. Let σ(A), W(A), and W0 (A) denote respectively the spectrum, the numerical range and the angular numerical range of A. For product of matrices, there are many inclusion results between the three sets. In particular, (i) if A is a multiple of a positive semidefinite matrix, then σ(AB) ⊆ W(A)W(B) for all n × n complex matrices B; (ii) if 0 6∈ W(A), then σ(AB) ⊆ W0 (A)W0 (B) for all n × n complex matrices B. In this thesis, we consider the converse of these results. For any fixed A, we find the necessary and sufficient conditions for the above inclusions to hold for all n × n complex matrix B. Some other inclusion relations between numerical range and angular numerical range are also considered.