Non-Linear Regression on a Matrix

Hi, I'm terribly sorry for submitting my primitive question, I'm a
beginner in R and was hoping to get some help re: non-linear fit.

I have a 2D data with the following structure:

A     B        C
1      1      111
1      2      121
1      3      131
2      1      141
2      2      151
2      3      161
3      1      171
3      2      181
3      3      191

I'm trying to fit C = non-linear function (A,B). I was wondering if
there's a package that would save my time of doing direct least square
estimation.

Thank you,
Diana