Matsuri is Yuji's current girlfriend, and a police officer as well. While she is oblivious to Yuji's attentions to Rio, she has noticed Rio as a threat to her marrying Yuji in the future, and has constantly threated Rio about staying away from him, obviously ignorant to the fact that Rio is not interested in Yuji, not to mention that Rio could possibly injure her if pushed too far.
Obviously, Matsuri seems to be a little more interested in being more "upwardly-mobile" in social stature because Yuji comes from a wealthy family. Ironically enough, her rivalry with Rio stems from believing Rio is using Yuji for her own benefit.
Source: http://en.wikipedia.org/wiki/Burn_Up_Scramble</a>