>>14120848It's because they are both a measure of volume.
If the vectors were independent they would measure a non zero volume. When they are dependent then the volume is 0 because one is in the same space as spanned by the others. This is also why nonzero determinant means invertible because in the case that your vectors represent basis vectors of a solution space to some equation in n variables, it is possible to find a unique solution to a system of equations when they span some nonzero volume. But if they are linearly dependent, the volume which is spanned by those vectors (the determinant) is 0, and there can only be trivial solutions because the volume of the solution space is 0.
