The determinant as area

How much bigger does a matrix make the plane? Watch the unit square grow into a parallelogram while a meter fills with its area.

How much bigger

The unit square has area 1. Under $\mathbf{A}$ it becomes the parallelogram with sides $(2,1)$ and $(-1,1)$, the columns of the matrix, as in A matrix moves the plane. What is its area?

Count it. Its corners are $(0,0)$, $(2,1)$, $(1,2)$ and $(-1,1)$. Draw the smallest upright rectangle around it: from $-1$ to $2$ across and $0$ to $2$ up, area $3\cdot2=6$. The rectangle’s four corners hold four right triangles outside the parallelogram, with areas $1$, $0.5$, $1$ and $0.5$. So the parallelogram’s area is $6-3=3$.

The matrix makes the unit square 3 times bigger. That factor is the determinant of $\mathbf{A}$, written $\det\mathbf{A}$. Here $\det\mathbf{A}=3$.

Every shape, the same factor

Not only the square grows by 3. Every small square of the grid becomes a copy of the same parallelogram, because a matrix moves every grid square alike. So every shape, built from many small squares, grows by the same factor. The circle of area $\pi$ becomes an ellipse of area $3\pi=9.425$. A triangle with corners $(0,0)$, $(1,0)$ and $(0,1)$, area $\tfrac12$, becomes the triangle on $(0,0)$, $(2,1)$ and $(-1,1)$, area $1.5$.

So $\det\mathbf{A}$ is one number that tells you how the matrix changes every area at once.

Sign and zero

After the clip, drag the second column. While it sits above the line of the first column, so that turning from $\mathbf{a}_1$ to $\mathbf{a}_2$ goes anticlockwise, the meter is positive. Drag it across the line of the first column, to $(2,0)$. The parallelogram now turns the other way, the letter F is mirrored, and the meter reads $-2$.

A negative determinant means the map flips the plane, like seeing it in a mirror. The size of the number is still the area factor; the sign tells you about the flip. Two different matrices can both scale area by 2, and the sign tells which one also mirrors.

Now drag the column onto the line of the first one, to $(4,2)$. The parallelogram is flat, the meter reads 0, and the whole plane lands on one line. A zero determinant means the map flattens the plane. Then no inverse can exist, as The inverse matrix showed: two points that land together cannot both be sent back.

The rules

Doing one map after another multiplies the area factors. First a shear, which slides the grid sideways without changing any area, then $\mathbf{A}$.

In symbols, with the shear first and $\mathbf{A}$ second, the product is $\mathbf{A}\mathbf{A}_\text{shear}=\begin{bmatrix}2&1\\1&2\end{bmatrix}$, and $\det(\mathbf{A}\mathbf{A}_\text{shear})=3=3\cdot1$. In general,

Three more rules follow.

• The inverse. $\mathbf{A}^{-1}$ undoes the growth, so $\det\mathbf{A}^{-1}=1/\det\mathbf{A}$. For our $\mathbf{A}$, $\tfrac13$.
• The transpose. $\det\mathbf{A}^\top=\det\mathbf{A}$. For our $\mathbf{A}$, $\mathbf{A}^\top=\begin{bmatrix}2&1\\-1&1\end{bmatrix}$ also has determinant 3.
• A multiple. $2\mathbf{A}$ doubles both columns, so the area grows by $2\cdot2=4$: $\det(2\mathbf{A})=12$. For an $n\times n$ matrix, $\det(c\,\mathbf{A})=c^n\det\mathbf{A}$.

Swapping the two columns turns the parallelogram the other way: $\begin{bmatrix}-1&2\\1&1\end{bmatrix}$ has determinant $-3$.

The formula

For a $2\times2$ matrix, the signed area of the parallelogram on the columns $(a_{11},a_{21})$ and $(a_{12},a_{22})$ is

It is the number that appeared in the inverse formula of The inverse matrix. For our $\mathbf{A}$: $2\cdot1-(-1)\cdot1=3$. With $\mathbf{a}_1=(2,1)$ fixed and $\mathbf{a}_2=(x,y)$, it is $2y-x$, the number the meter shows while you drag.

In three dimensions, the determinant is the signed volume of the box on the three columns. The columns $(1,1,0)$, $(1,0,1)$ and $(0,1,1)$ make a box of volume 2, turned inside out, so the determinant is $-2$. Computing determinants shows how to get it.

Worked example

1. The matrix of the page. $\det\begin{bmatrix}2&-1\\1&1\end{bmatrix}=3$; the half-unit triangle becomes a triangle of area $1.5$.
2. A product. $\det(\mathbf{A}\mathbf{A}_\text{shear})=\det\begin{bmatrix}2&1\\1&2\end{bmatrix}=4-1=3=3\cdot1$.
3. Other maps. Scaling by 1.5 in every direction: $2.25$. The reflection $\begin{bmatrix}0&1\\1&0\end{bmatrix}$: $-1$. The projection $\begin{bmatrix}0.5&0.5\\0.5&0.5\end{bmatrix}$: $0.25-0.25=0$. Any rotation or shear: $1$.
4. A stretch. $\det\begin{bmatrix}3&0\\0&2\end{bmatrix}=6$.
5. A flat one. $\det\begin{bmatrix}1&2\\2&4\end{bmatrix}=0$: the columns lie on one line.
6. Rules. If $\det\mathbf{A}=3$ and $\det\mathbf{B}=-2$, then $\det(\mathbf{A}\mathbf{B})=-6$.
7. Predict. As $\mathbf{a}_2$ crosses the line of $\mathbf{a}_1$, the sign changes: the plane flips.

Where you’ll meet this

A graphics engine checks the sign of each object’s matrix. A negative determinant means the object has been mirrored, so its triangles now wind the other way round, and the engine must flip them to draw their front faces.

When a change of variables stretches a region, every area is multiplied by a determinant. Polar coordinates stretch a small patch by the distance $r$ from the centre, so at $r=2$ areas are doubled. A probability density moved by a map must shrink by the same factor to keep its total at 1.

Reference card