Let be a matrix if you separate it into its column vectors, and call them and , then you can plot and in the -plane, and create a parallelogram using them as two of its adjacent sides.

The area of that parallelogram is simply the absolute value of the determinant of the matrix .

paralellogram xy-plane of vectors


Let be a matrix that defines a figure in the -plane, be a transformation matrix applied to and - the new figure derived from the transformation of . We know the area of will be .

But we also can describe the area of in terms of and the transformation matrix :