No matter which row you expand, you should always get the same determinant. So you must've made a mistake when expanding across row 2.
As for matrices in general, of course you can have a non-positive determinant. Consider the matrix:
I got the matrix A
3 2 -1
1 6 3
2 -4 0
I'm not sure if determinants are always positive, but they must be because in my textbook, the determinant is 64.
I got this calculating the minors of row 1. I got 36+12+16. For row 2 I got -4-12-48 and row 3 I got 24-40+0.
This leads me to believe dets are always positive, but I'm not sure. What are the conditions for the sign of determinants?
Expanding row 2:
(-1)^2+1 * 1 det (2*0)-(-1*-4) = 4
+
(-1)^2+2 * 6 det (3*0)-(-1*2) = 12
+
(-1)^2+3 * 3 det (3*-4)-(2*2) = 48
Okay I get 64 as well. I understand my mistake. Originally I did not pay attention to the sign and sign changes depending on the location of the minors 4, 12, 48. So originally I had -4, -12 and -48 because I forgot to change the sign, or apply the correct ij in the (-1)^ij part.