Find the area of the triangle with vertices (2, 3), (-1, 0) and (4, -2).

The area of the triangle with vertices $(x_i, y_i)$ is $\tfrac{1}{2}\left|\det\right|$ of the matrix with rows $(x_i, y_i, 1)$. Three points are collinear exactly when this determinant is $0$. A matrix multiplies every area by $|\det T|$.