Matrices as transformations: matrix times vector, determinant and area

Warm-up

Three quick questions from earlier lessons.

Discover

Doubling both coordinates of every point of a square with area 1 gives a square with area…

4
2

Key idea

A matrix moves vectors; its determinant scales areas.

Multiplying a column vector by a matrix gives a new vector:

\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}

. So a

2 \times 2

matrix is a transformation of the plane; its columns are the images of

(1, 0)

and

(0, 1)

. Areas are multiplied by

|\det A|

.

Worked example

Apply $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ to (3, 0) and (0, 3).

(0 · 3 − 1 · 0, 1 · 3 + 0) = (0, 3)

First point.

(−3, 0)

Second point.

a 90° counterclockwise rotation; det = 1, areas unchanged

Conclusion.

Your turn, guided

Your turn: the area scale factor of $\begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}$ .

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det = 3 · 2.

Practice ★

Let A = $\begin{pmatrix} 9 & 2 \\ 3 & 6 \end{pmatrix}$ . What is the first component of A $\begin{pmatrix} -3 \\ -1 \end{pmatrix}$ ?
Let A = $\begin{pmatrix} -1 & 5 \\ -5 & -5 \end{pmatrix}$ . What is the first component of A $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ ?

The matrix $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ maps the point (8, 0) to (0, 8) and (0, 8) to (−8, 0). Which transformation does it represent?
1. A rotation of 90° counterclockwise about the origin
2. A reflection in the line y = x
3. A rotation of 90° clockwise about the origin

Watch out!

Practice ★★

A shape with an area of 2 square units is transformed by the matrix $\begin{pmatrix} 2 & 2 \\ 2 & 3 \end{pmatrix}$ . What is the area of the image?
A shape with an area of 8 square units is transformed by the matrix $\begin{pmatrix} 2 & 2 \\ 1 & 3 \end{pmatrix}$ . What is the area of the image?

Let A = $\begin{pmatrix} 9 & 2 \\ 3 & 6 \end{pmatrix}$ . What is the first component of A $\begin{pmatrix} -3 \\ -1 \end{pmatrix}$ ?

Exit ticket

Three questions to check you have got it.

Let A = $\begin{pmatrix} 9 & 2 \\ 3 & 6 \end{pmatrix}$ . What is the first component of A $\begin{pmatrix} -3 \\ -1 \end{pmatrix}$ ?
Let A = $\begin{pmatrix} -1 & 5 \\ -5 & -5 \end{pmatrix}$ . What is the first component of A $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ ?
Let A = $\begin{pmatrix} 7 & -1 \\ -2 & 7 \end{pmatrix}$ . What is the first component of A $\begin{pmatrix} 6 \\ 1 \end{pmatrix}$ ?

Summary card

Matrices as transformations

Av moves vectors; columns are the images of (1, 0) and (0, 1); areas scale by |det|.

Watch out: the area factor is the determinant, not a sum of entries.

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