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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.

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    $\begingroup$ Note $\ $ Pick employed this and his area theorem to give a beautiful geometric proof of the Bezout linear representation of the GCD. $\endgroup$
    – Bill Dubuque
    Mar 26, 2011 at 15:45
  • $\begingroup$ Have a look at this. $\endgroup$
    – Giuseppe Negro
    Mar 26, 2011 at 16:25
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    $\begingroup$ See if your question is answered by the discussion here: math.stackexchange.com/questions/668/… . A short answer is that this should be taken (properly modified to take orientation into account) as the definition of the determinant. $\endgroup$
    – Qiaochu Yuan
    Mar 26, 2011 at 16:46
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    $\begingroup$ n.b. as pointed out by @martin-sleziak in his answer below, (3) and (4) are not (generally) parallelograms $\endgroup$
    – JobJob
    Dec 25, 2015 at 11:42
  • $\begingroup$ Going back to the determinant view of this, it might be useful for people to note that 1 and 2 can be seen to have the same area from the fact that det(A)=det(transpose(A)) $\endgroup$
    – JasoonS
    Jun 15, 2016 at 12:44

11 Answers 11

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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.)

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  • $\begingroup$ Can you point to the actual book this was taken from? I'd like to explore this further. = ) $\endgroup$
    – Phonon
    Jun 6, 2014 at 0:46
  • $\begingroup$ Image link is broken for me. Here's a re-paste: i.stack.imgur.com/gCaz3.png $\endgroup$
    – Laser
    Aug 16, 2015 at 5:52
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enter image description here

I know I'm extremely late with my answer, but there's a pretty straightforward geometrical approach to explaining it. I'm surprised no one has mentioned it. It does have a shortcoming though - it does not explain why area flips the sign, because there's no such thing as negative area in geometry, just like you can't have a negative amount of apples(unless you are economics major).

It's basically:

  Parallelogram = Rectangle - Extra Stuff.

If you simplify $(c+a)(b+d)-2ad-cd-ab$ you will get $ad-bc$.

Also interesting to note that if you swap vectors places then you get a negative(opposite of what $ad-bc$ would produce) area, which is basically:

  -Parallelogram = Rectangle - (2*Rectangle - Extra Stuff)

Or more concretely:

$(c+a)(b+d) - [2*(c+a)(b+d) - (2ad+cd+ab)]$

Also it's $bc-ad$, when simplified.

The sad thing is that there's no good geometrical reason why the sign flips, you will have to turn to linear algebra to understand that.

Like others had noted, determinant is the scale factor of linear transformation, so a negative scale factor indicates a reflection.

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  • $\begingroup$ davidroodman.com/blog/2016/02/21/… $\endgroup$
    – bfhaha
    Jul 31, 2020 at 6:04
  • $\begingroup$ "If you simplify $(c+a)(b+d)−2ad−cd−ab$ you will get $ad−bc$" no, this is $bc - ad$ $\endgroup$
    – user50229
    Jun 9, 2022 at 16:49
  • $\begingroup$ The second equation is also wrong, $(c+a)(b+d)−[2∗(c+a)(b+d)−(2ad+cd+ab)] = ad - bc$ $\endgroup$
    – user50229
    Jun 9, 2022 at 16:55
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The oriented area $A(u,v)$ of the parallelogram spanned by vectors $u,v$ is bilinear (eg. $A(u+v,w)=A(u,w)+A(v,w)$ can be seen by adding and removing a triangle) and skew-symmetric. Hence $A(ae_1+be_2,ce_1+de_2)=(ad-bc)A(e_1,e_2)=ad-bc$. (the same works for oriented volumes in any dimension)

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For the matrix $\left[\begin{array}{cc} a & c \\ b & d \\ \end{array}\right]$ let $$A = \left[\begin{array}{c} a \\ b \\ \end{array}\right] \;\text{and}\; B = \left[\begin{array}{c} c \\ d \\ \end{array}\right]$$

as shown in the following figure.

Determinant Figure

Then the height of the parallelogram is

$$\text{height} = |B|\sin\alpha = |B|\cos\beta.$$

If we rotate $A$ by 90 degrees in the CCW direction as follows:

$$R_{90º}A = \left[\begin{array}{cc} 0 &-1 \\ 1 &0 \\ \end{array}\right] \left[\begin{array}{c} a \\ b \\ \end{array}\right] = \left[\begin{array}{c} -b \\ a \\ \end{array}\right],$$

maintaining the magnitude of the base as

$$\text{base} = |A| = |R_{90º}A|,$$

then it is clear that the area of the parallelogram is therefore

$$ \text{base}\times\text{height}=(|A|)(|B|\sin\alpha) = |R_{90º}A|\;|B|\cos\beta = (R_{90º}A)\cdot B = \left[\begin{array}{c} -b \\ a \\ \end{array}\right] \cdot \left[\begin{array}{c} c \\ d \\ \end{array}\right] = ad-bc. $$ Q.E.D.

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    $\begingroup$ +1. While it's clear that $|A|\sin\alpha=|R_{90^\circ}A|\cos\beta$, it may be visibly clearER in your diagram that $|\text{altitude of parallelogram}|=|B|\sin\alpha=|B|\cos\beta$ ... which ---since $|A|=|R_{90^\circ}A|$--- works just as well in your argument. (Drawing the altitude from the tip of $B$ onto $A$ would help drive this home.) $\endgroup$
    – Blue
    Mar 13, 2012 at 3:29
  • $\begingroup$ @Don, good point. I will make the edit tonight. $\endgroup$
    – Tpofofn
    Mar 13, 2012 at 10:27
  • $\begingroup$ I posed a video on youtube explaining this answer. youtu.be/m0DrWT18gBA $\endgroup$
    – Tpofofn
    Aug 4, 2017 at 2:23
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I keep forgetting the tricks to prove this so I found the only way to remember it is to stick to basic principles. We can deform the parallelogram to get a square like so. enter image description here

Now we only need to find the side lengths of this rectangle. The height is easy, it's just d. To get the base length, we find:

enter image description here

and so with a little algebra we find that the base length is $a-b\times\frac{c}{d}$.

Hence the area is just $$ d(a-\frac{bc}{d}) = ad - bc$$

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  • $\begingroup$ Very nice answer! Thank you. $\endgroup$
    – Mahathi Vempati
    Jun 6, 2017 at 4:46
  • $\begingroup$ I did not get the part where you said, 'with a little algebra'. Could you please explain your steps how you come to the fact that the base length of your rectangle equals: $a-b*c/d$? $\endgroup$
    – Mark
    Feb 17, 2021 at 1:45
  • $\begingroup$ What is the length of the horizontal green line? $\endgroup$
    – samlaf
    Feb 17, 2021 at 1:47
  • $\begingroup$ The base length is $a - x$, where $x$ is the basis of the green triangle. That green triangle has height $b$ and a opposite angle $\alpha$. Then $$x = b/\tan(\alpha)$$ But note that $\alpha$ is the angle of the vector $(c,d)$ with the x-asix, then $\tan(\alpha) = d/c$. Hence $$x = b*c/d$$ The result follows $\endgroup$
    – Astrid A. Olave H.
    May 31 at 15:12
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Here is an animated gif I made to derive the area of a parallelogram.

wordless proof about area of parallelogram

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  • $\begingroup$ That's so cool! $\endgroup$
    – David Callanan
    Jan 10, 2022 at 11:03
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If you compute the cross product of (a,b,0) and (c,d,0), then you get (in the third coordinate) ad-bc. This is, up to the sign, the area of the parallelogram.

BTW I think that (3) and (4) are not parallelograms, are they?

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  • $\begingroup$ Thank you. (3) and (4) are not parallelograms. $\endgroup$
    – user
    Mar 26, 2011 at 16:56
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Also, if the coordinates of any shape are transformed by a matrix, the area will be changed by a scale factor equal to the determinant.

Since the determinant is the scale factor when the unit square is transformed to a parallelogram, it will be the scale factor when any parallelogram with the origin as a vertex is transformed to any other parallelogram because the inverse matrix will transform a parallelogram back into a square and has reciprocal determinant. If there is no inverse, the determinant is 0 and the transformed shape has no area.

Any triangle with the origin as a vertex can be drawn as half of a parallelgram including the origin. Any triangle not including the origin is the area of a triangle containing the origin minus two triangles inside not containing the origin. The area of any shape can be split into triangles, although an infinite number will be required if it has curved sides.

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Wrote this for a linear algebra class of mine. The argument is predicated on using shears.

Assume you have two vectors, (a, ay) and (xd, xyd+d). Weird choice and abundance of variables to be explained in a moment. I can obviously find the determinant of this, which is ad (do it).

So if I want to prove that the determinant is an area, I need to show that these weirdo vectors share an area with (a,0) and (0,d), which also has the determinant ad. Well, it turns out what I can do is shear the matrix with (a,0) and (0,d) as columns, since a shear does not alter the area at all (show this geometrically).

Take vector (a,0) and (0,d) and apply shear matrix (1,x,0,1), followed by (1,0,y,1), which gives you the original weirdo vectors (a,ay) and (xd, xyd+d). Since the shears do not change area, and we know the area of the rectangle formed by (a,0) and (0,d), the area of two arbitrary vectors may be expressed by its determinant, which we have shown to be identical to the determinant of rectangular matrix (a,0,0,d). QED.

You can extend this argument to 3D, etc. by applying shears in any arbitrary direction. Therefore, a determinant will provide you with a volume in any R^n.

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I made the following slide to explain where this formula comes from:

enter image description here

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Another geometric proof using an animation in desmos:

The green rectangle and parallelogram have the same base and height:

$\operatorname{green}=\operatorname{area}\left((0,0)(a,0)(a,d)(0,d)\right)=ad= \operatorname{area}\left((0,0)(a,0)(a,{c}+d)(c,d)\right).$

The orange rectangle and parallelogram have the same base and height:

$\operatorname{orange}=\operatorname{area}\left((a,d)(a+c,d)(a+c,b+d)(a,b+d)\right)=bc= \operatorname{area}\left((a,0)(a+c,d)(a+c,b+d)(a,b)\right).$

The violet, green and orange parallelograms have the same bases, the height of green parallelogram is the sum of heights of violet and orange parallelograms.

$\operatorname{violet} + \operatorname{orange} = \operatorname{green}$

$\operatorname{violet}=\operatorname{area}\left((0,0)(a,b)(a+c,b+d)(c,d)\right) = \operatorname{green} - \operatorname{orange} = ad-bc$

enter image description here

enter image description here

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