Do PS3s play PS2 games?

I read that the European model is backwards compatible but the American model isn’t. When I rang and asked at the store I bought mine he said it would probably be American. However, I found that I could play my old PS1 Games by creating an internal PS1 memory card. I therefore assumed my model was European. I assumed I could also play PS2 games because it had an option to make a PS2 internal memory card BUT, I went to put in a friends PS2 game, and it said something like (can’t remember exactly sorry) “The PS3 is not formatted to play PS2 games”. I am going to try some of my friend’s other games to see if it was just maybe that one game but I won’t be able to for a few days and I was just wondering if anyone could tell me.