In my book on linear algebra (first year university) the author starts off explaining the concepts of determinants by focusing on dimensions n<=3, giving geometrical definitions of area and volume. Then he goes on to explain that if the area/volume given by the columns vectors is zero obviously this means there is linear dependance in these columns and thus det A=0 is equivalent to AX=0 having non-trivial solutions.
I'm with him so far. But moving into higher dimensions, he suddenly gets very ... vague... So I've googled the net, looked at wikikpedia, but I still haven't really understood for n>3 what I understood for n<=3. Wikipedia gives ways to calculate determinants for n>3, but no real definition, at least not anything where the important implications of the determinant being zero or not have for systems of equation become clear.
If anyone could help me with this I'd be very happy! In summary:
1. Definition of determinant for any n.
2. Validity of implications of Det A=0 or Det A<>0 true for n<=3 also for n>3.