I know the concept. I know the geometric explanation of the transformed volume. But the determinant still troubles me. The geometric explanation does not help when we are solving a non-vector problem, except that it could be assumed that the most different the columns in a matrix are from each other the greater the determinant, as the cube gets a higher volume when the vectors that form it are far away. That brings to the question: what really does the determinant depend on (ie. how large the numbers in the matrix are or how much the differ from each other)?

In my mind, the answer to this question should lead to an intuitive idea of the determinant different from the geometrical explanation.

Many thanks in advance.