Matrices and Transformations

Syllabus tag: KCSE | Mathematics | Form 4 | Topic 1 Matrices and Transformations

Lesson objectives

By the end of this topic, you should be able to:

• Relate image and object under a given transformation on the Cartesian plane.
• Determine the matrix of a transformation.
• Perform successive transformations and find the single equivalent matrix.
• Use the determinant to find the area scale factor and identify inverse transformations.

Matrices and Transformations

A transformation on the Cartesian plane can be written as a matrix, and that turns geometry into algebra.

a) How it works

Write a point as a column vector.

Multiplying by the transformation matrix gives the image point.

For several points, place them side by side as columns in one matrix and multiply once. That transforms the whole shape in a single step.

b) The standard matrices

Each of these describes a transformation about the origin. A rotation about any other centre needs a different treatment.

c) Finding the matrix of a transformation

The columns of the matrix are the images of the unit vectors.

The first column is the image of (1, 0). The second is the image of (0, 1).

So to find any transformation matrix, work out where those two points go. Write the results as columns.

This is quicker and more reliable than solving simultaneous equations from a general point.

d) Successive transformations

Applying transformation A then transformation B is represented by the product BA, not AB.

The order reverses because the first transformation applied sits nearest the point vector, which is on the right.

That reversal is the standard error in this topic. Write the vector on the right and build outwards.

The single matrix BA is the equivalent single transformation.

e) The determinant as area scale factor

The determinant of the transformation matrix is the area scale factor.

So a matrix with determinant 3 turns a shape of area 5 into one of area 15.

A negative determinant means the orientation is reversed, as in a reflection. Take the absolute value for the area itself.

A determinant of 1 means area is preserved. Rotations and reflections both have determinant ±1, which is why they produce congruent images.

A determinant of zero means the shape collapses onto a line, and the transformation cannot be undone.

f) Inverse transformations

The inverse matrix undoes the transformation.

A⁻¹ maps every image back to its object.

A matrix with determinant zero is singular and has no inverse. That matches the fact that a collapsed shape cannot be restored.

The inverse of a rotation through θ is a rotation through −θ. The inverse of a reflection is that same reflection, since reflecting twice returns the original.

g) Where this is used

Computer graphics, where every rotation and scaling is a matrix. Robotics. Image processing. Any repeated geometric operation.

Words to know

• Transformation matrix -- a 2 × 2 matrix mapping points of the plane to their images.
• Unit vectors -- the points (1, 0) and (0, 1), whose images form the matrix columns.
• Area scale factor -- the factor by which areas are multiplied, equal to |det|.
• Singular transformation -- one with determinant zero, collapsing the plane onto a line.
• Successive transformations -- two or more applied in order.

:::checkpoint Check yourself

1. Write the matrix for a reflection in the x-axis.
2. Find the image of (3, 2) under the matrix (0 −1 / 1 0).
3. A transformation has determinant −2. What happens to a shape of area 6?
4. Why does applying A then B give the matrix BA? :::

Bridge to practice

The exercises begin with applying matrices to points and finding matrices from unit vectors, move through the standard transformations and the determinant, and finish with successive transformations and inverses. For every combined transformation, write down which acts first before forming the product.