TITLE: Parallelogram area using determinant
QUESTION [5 upvotes]: Given a Parallelogram with the co-ordinates: $(a+c, b+d), (c,d), (a, b)$ and $(0, 0)$
I have to prove that the area of the Parallelogram is:
$|ad-bc|$ as in the determinant of:
$$\begin{bmatrix}
a & b\\ 
c & d
\end{bmatrix}$$
How do I even begin using the concept of determinants for this geometrical question?

REPLY [5 votes]: Firstly, show that the transformation of the points of the unit square map to the parallelogram that you show.  Secondly, calculate the area of a parallelogram using some basic symmetries of the shape and show it is $|a d - b c|$.  This is in fact the basic principle behind determinants, they were invented to see how the area of shapes change under a matrix transformation.
How you show the area depends on what you already know.  If you know about complex numbers, think about what the imaginary part of $\bar{z_1} z_2$ represents.