WHY MATSUDA?! Seriously, he could easily have Aizawa do that... What's your vendetta against Matsu? X< And besides, it'd be out of character for a sweetheart like Matsuda do that. Someone would know something's up.
Didn't te watch the anime? This website asks a bunch of stupid questions. Once they meet L, L gives them all fake names and tells them to never disclose their real name to anyone, not even those closest to them until the Kira case is over. Light doesn't know anyone's real name except for his father and he's not about to kill his father. He got his father killed in the end because he felt he had no other choice.
I know I'm late in this but i'll answer anyway. I dont think that's possible o else Light could have killed everyone very easily. This would be because Light has to know a person's real and full name and write that in his Death Note in order to kill that individual and in this case he doesnt know L's real name to actually kill him. So,"Matsuda will ask a domanda in private of L and then shoot him" wouldnt work because Light didnt write L's actual name. This L is not his full name, if not real.