Matrices and Transformations

Slide 1 · Learning Outcomes

What you will be able to do

In Form 2 you drew reflections, rotations and enlargements on graph paper. In Form 4 a 2 × 2 matrix does the same job: multiply it by the coordinates of a point and you get the image.

Find the image of a point or shape under a transformation matrix.
Find the matrix of a transformation using the unit square or given images.
Identify the matrices of common reflections, rotations and enlargements.
Find a single matrix for successive transformations.
Use inverse matrices to find an object from its image.
Use the determinant as the area scale factor.
Describe shears and stretches, and tell isometric from non-isometric transformations.

The key idea: abcd × xy = ax + bycx + dy

Slide 2 · Images under a matrix

Matrix playground

Type any matrix. The green triangle is the object A(1, 1), B(3, 1), C(1, 2). The orange triangle is its image.

M =

objectimage

Tip: write the object's coordinates as columns of one matrix: 131112. One multiplication then gives all the image points at once.

Slide 3 · Finding the matrix

The unit square method

The images of I(1, 0) and J(0, 1) give the two columns of the matrix directly:

If I(1,0) → (a, c) and J(0,1) → (b, d), then M = abcd

Activity: find the matrix

M =

Other images? If you are given images of other points, write M = abcd, multiply out, and solve the simultaneous equations for a, b, c and d.

Slide 4 · Common transformation matrices

Know these by heart

Tap a transformation.

All rotations and enlargements here are about the origin. Positive angles are anticlockwise. In general, rotation through θ: cos θ−sin θsin θcos θ

Slide 5 · Successive transformations

One after another

If transformation A is followed by transformation B, the single matrix is BA (the first one goes on the right, next to the point).

First (A)

Then (B)

objectafter Aafter B

Slide 6 · Inverse and area scale factor

Undoing a transformation

Inverse: M⁻¹ maps the image back to the object.
For M = abcd, M⁻¹ = ¹⁄_{(ad − bc)} d−b−ca

Activity: determinant and inverse

M =

Slide 7 · Shear and stretch

Shear and stretch

Shear

Points slide parallel to an invariant line, by an amount proportional to their distance from it. With the x-axis invariant: 1k01. Area is unchanged (det = 1).

Stretch

Points move perpendicular to an invariant line, distances multiplied by k. With the y-axis invariant (stretch parallel to the x-axis): k001. Area × k.

k2

Slide 8 · Isometric and non-isometric

Does the shape keep its size?

An isometric transformation keeps lengths, angles and area: the image is congruent to the object. For an isometric matrix, |det| = 1 and lengths are preserved.

Activity: sort the transformations

Tap a card, then tap the box it belongs to.

Isometric (congruent image)

Non-isometric

Watch out: a shear has det = 1 (area kept) but it is still non-isometric, because lengths and angles change.

Slide 9 · Practice

Worked example and quick checks

Worked example

A transformation maps (1, 0) to (2, 1) and (0, 1) to (−1, 3). (a) Find its matrix M. (b) A triangle of area 4 cm² is transformed by M. Find the area of the image.

Step 1: The images of I and J are the columns: M = 2−113

Step 2: det M = (2)(3) − (−1)(1) = 6 + 1 = 7

Step 3: Image area = |det M| × object area = 7 × 4

Answer: 28 cm²

Your turn

1. The image of (3, −2) under 0110 is (, ).

2. det 4325 =

3. A square of area 3 cm² is enlarged by 3003. Image area = cm².

4. Reflection in the x-axis followed by reflection in the y-axis is equivalent to a rotation of about the origin.

Slide 10 · Quiz

Check your understanding

Part A: Fill in the blanks (auto-marked, 1 mark each)

1. 0−110 represents a rotation of + about the origin.

2. det 3124 =

3. The area scale factor of 2002 is

4. The image of (2, 3) after reflection in the x-axis is (2, ).

5. Under a shear, the area of a shape remains .

6. A transformation that keeps the size and shape of an object is called .

7. k00k represents an with centre the origin.

8. If transformation A is followed by B, the single matrix is .

9. The inverse of a reflection in the line y = x is a reflection in the line y = .

10. A matrix with determinant has no inverse.

Part B: Short answers (smart-marked against key points)

Masomotele · Masomotele Technical Training Institute (MTTI), Eldoret · Free KCSE revision