Why determinant of a 2 by 2 matrix is the area of a parallelogram?

Question 1

Let $A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}$ .

How could we show that $ad-bc$ is the area of a parallelogram with vertices $(0, 0),\ (a, b),\ (c, d),\ (a+b, c+d)$ ?

Are the areas of the following parallelograms the same?

$(1)$ parallelogram with vertices $(0, 0),\ (a, b),\ (c, d),\ (a+c, b+d)$ .

$(2)$ parallelogram with vertices $(0, 0),\ (a, c),\ (b, d),\ (a+b, c+d)$ .

$(3)$ parallelogram with vertices $(0, 0),\ (a, b),\ (c, d),\ (a+d, b+c)$ .

$(4)$ parallelogram with vertices $(0, 0),\ (a, c),\ (b, d),\ (a+d, b+c)$ .

Thank you very much.

Question 2

Spend a little time with this figure due to Solomon W. Golomb and enlightenment is not far off:

enter image description here

(Appeared in Mathematics Magazine, March 1985.)

Answer