The Determinant from Signed Volume to the Laplace Expansion

We present a novel perspective on developing the determinant through the lens of signed volume. Starting with a unique and rigorous development of both the volume and orientation of a parallelepiped, we are able to give an unambiguous, basis-free definition for the determinant of a linear transformation. We then build intuition for the determinant by proving many of its properties in a succinct and basis-free fashion. We conclude our journey by using these properties to derive a well-known method for computing the determinant and motivating the Laplace expansion.