This is the fourth lecture in our series on Linear Algebra. Here we introduce area as a determinant, first in two dimensions, then in three. We give a pictorial definition using the affine grid plane, then also a purely algebraic approach using Grassmann's bi-vectors. A bi-vector is a two dimensional analog of a vector, and captures the physical notions of torque, angular momentum, and force on a charged particle in a magnetic field.

We find the classical formulas for 2 and 3 dimensional determinants.