,     ,        

Then ,   

           det(A) = det(B) + det(C).

                    P(u+v,w) = Area of the parallelogram with sides u+v and w.                                          

                    P(u,w) = Area of the parallelogram with sides u.                                                     

                    P(v,w) = Area of the parallelogram with sides v, w.                                               

                    then ,  P(u+v,w) = P(u,w) + P(v,w)  -----------------(*)                                          

since , Area (OBD) = Area (AQR) ,

In the applet below , you can click at any three points to find u, v, w .Click the "Show" button to see value of these vectors .

Click on the  "Show det(A(u+v,w)) = det(A(u,w)) + det(A(v,w))" to see that the required property (*) hold.