Yes, C maj scale has the same notes A min (aeolian mode) has, but the only difference is the root, which will give you different paterns if you changed the root on the same string.
It's very easy to prove this on a piano or a key board, the A minor scale has no black keys, just the white keys, exaclty like the C maj.
A min : A B C D E F G (A)
C maj : C D E F G A B (C)
All of the above notes are white keys on the piano (key board) since there ain't any flats or sharps.
