Matrices as transformations - what a matrix does to the plane, step by step

The idea: a matrix moves the whole plane

A matrix is a table of numbers. A $2 \times 2$ matrix has four of them, in two rows and two columns:

You can read it as a recipe for moving points: the point $(x, y)$ goes to the point $(ax + by,\ cx + dy)$. Every point of the plane moves at the same time, so the whole plane is transformed. It can turn, stretch, slant or flip over.

The recipe is simpler than it looks. You only need to know where two arrows go: $e_1 = (1, 0)$, one step along the $x$-axis, and $e_2 = (0, 1)$, one step up. Their new positions are the two columns of the matrix, and the rest of the grid follows.

Watch the grid change

Go through the animation step by step with the buttons or the arrow keys. The faint grid is the plane before the move, and the coloured grid shows where the matrix takes it. The arrows $Me_1$ and $Me_2$ are the images of $e_1$ and $e_2$ - where the matrix $M$ sends them.

Two arrows decide everything. Every point can be built from them, $(x, y) = x\,e_1 + y\,e_2$, so once you know where $e_1$ and $e_2$ go, you know where every point goes.

Try it yourself: build a matrix

Set the four numbers of the matrix with the sliders. The left pair is the first column - where $e_1$ goes. The right pair is the second column - where $e_2$ goes. Watch the grid, the square and the flag, and read the determinant under the sliders.

What to notice:

• the rotation by $90°$ and the shear change the square’s position or shape, but not its area: both have determinant $1$,
• the stretch $\begin{pmatrix} 2 & 0 \\ 0 & 1.5 \end{pmatrix}$ multiplies every area by its determinant, $3$,
• the reflection $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ has determinant $-1$: the area stays $1$, but the flag is mirrored,
• with parallel columns, such as $(1, 0.5)$ and $(2, 1)$, the determinant is $0$ and the plane is flattened onto a line,
• drag one column across the other: just as they line up, the determinant passes through $0$, changes sign, and the flag flips over.

The definition

A matrix as a transformation. The matrix $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ sends the point (or vector) $(x, y)$ to

Each new coordinate is a row of the matrix times the vector: multiply the matching numbers and add them up.

The columns are the images of e₁ and e₂. Put in $(1, 0)$ and you get $(a, c)$, the first column. Put in $(0, 1)$ and you get $(b, d)$, the second. Every vector is $x\,e_1 + y\,e_2$, so

To write down the matrix of a transformation, find where $e_1$ and $e_2$ go and write their images as columns.

Linearity. Such a transformation is linear: $M(\mathbf{u} + \mathbf{v}) = M\mathbf{u} + M\mathbf{v}$ and $M(k\mathbf{u}) = k\,M\mathbf{u}$ for every number $k$. So the origin stays put. A line through the point $\mathbf{p}$ in the direction $\mathbf{d}$ is made of the points $\mathbf{p} + t\mathbf{d}$, and they go to the points $M\mathbf{p} + t\,M\mathbf{d}$. That is a line in the direction $M\mathbf{d}$, or a single point when $M\mathbf{d} = \mathbf{0}$. Parallel lines share their direction, so their images are parallel too - unless the whole plane is squashed onto one line.

The determinant. The number

is the determinant of $M$. Its absolute value $|\det M|$ is the area of the image of the unit square - the parallelogram spanned by the two columns. Every other area is multiplied by the same number, so $|\det M|$ is the area scale factor: a figure of area $S$ goes to a figure of area $|\det M| \cdot S$. The sign says more:

• $\det M > 0$: figures keep their orientation,
• $\det M < 0$: figures are mirrored,
• $\det M = 0$: the columns are parallel and the plane is squashed onto a line or a point. Such a transformation cannot be undone - $M$ has no inverse.

One transformation after another. Doing $A$ first and then $B$ is the transformation with matrix $BA$ - the one done first stands on the right. The order matters: usually $BA \ne AB$. First $A$ scales areas, then $B$ does, so $\det(BA) = \det B \cdot \det A$.

Example 1: the image of a triangle and its area

The matrix $M = \begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix}$ moves the triangle with corners $O = (0, 0)$, $A = (2, 0)$ and $B = (0, 1)$. Where does the triangle go, and what is its new area?

Example 2: the matrix of a rotation by 90°

Find the matrix of the rotation through $90°$ anticlockwise about the origin.

Your turn

A transformation sends $e_1$ to $(1, 2)$ and $e_2$ to $(-1, 1)$. Write down its matrix, find the image of the point $(2, 3)$ and say what the transformation does to areas.

Common mistakes

• Rows instead of columns. The images of $e_1$ and $e_2$ are the columns of the matrix. Written as rows, the rotation $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ becomes $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ - a rotation the other way.
• “The order does not matter.” Take $R$, the rotation by $90°$ anticlockwise, and $F = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$, the reflection in the $x$-axis. Rotating first and then reflecting gives $FR = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$, the reflection in the line $y = -x$. Reflecting first gives $RF = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, the reflection in $y = x$.
• “A negative determinant means a negative area.” An area is never negative. When $\det M = -2$, every area doubles and figures are mirrored.
• “A determinant of 0 means a matrix full of zeros.” The matrix $\begin{pmatrix} 2 & 1 \\ 1 & 0.5 \end{pmatrix}$ has no zero in it, yet its determinant is $2 \cdot 0.5 - 1 \cdot 1 = 0$. Its columns are parallel, so it flattens the plane onto a line.

In exams and at university

In England, matrices are not part of A-level Mathematics, but they belong to the core content of A-level Further Mathematics that every exam board must include. Students use $2 \times 2$ matrices for transformations of the plane, such as rotations, reflections, stretches and enlargements. They combine two transformations by multiplying their matrices, with the same rule as on this page: $AB$ means $B$ first, then $A$. They also interpret the determinant as the area scale factor, including its effect on orientation, and find invariant points and lines - points and lines that the transformation maps onto themselves. The unit vectors that this page calls $e_1$ and $e_2$ are written $\mathbf{i}$ and $\mathbf{j}$ at A-level.

In the United States, AP Precalculus ends with the unit Functions Involving Parameters, Vectors, and Matrices. In it, a $2 \times 2$ matrix acts as a linear transformation of the plane, and the absolute value of its determinant shows how areas are scaled. That unit is not tested on the AP exam: teachers may include it, depending on state or local requirements.

At university, matrix transformations and determinants belong to a first course in linear algebra. In the open textbook Understanding Linear Algebra by David Austin, matrix transformations come in the second chapter, a few sections after the product of a matrix and a vector. The determinant follows in the third chapter, where it is defined as a signed area - the same picture as on this page.

Check yourself

Flashcards for review

Flip a card and say whether you knew the answer. The ones you did not know come back at the end of the deck. Why it pays to come back to them in a few days is explained in the guide Spaced repetition.