What is the determinant of a matrix?
The determinant is basically the factor by which a matrix scales the area/volume/hypervolume - with negative value if reflection is involved. If the result is zero then the matrix collapses
For example, in
#abs((a,b),(c,d)) = ad - bc#
#abs((a_11,a_12,a_13),(a_21,a_22,a_23),(a_31,a_32,a_33)) =a_11 abs((a_22,a_23),(a_32,a_33)) + a_12 abs((a_23,a_21),(a_33,a_31)) + a_13 abs((a_21,a_22),(a_31,a_32))#
A similar process can be used on larger matrices, but there are quicker methods.