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

Let A=[abcd].

How could we show that adbc 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.

Answer

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

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(Appeared in Mathematics Magazine, March 1985.)

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