Introduction to Matrix Transformations

In our previous chapters, we learned how to add and multiply matrices. Now, we are going to use those skills for something much more visual! Imagine a shape on a grid. A matrix transformation is like a mathematical "instruction" that tells every point on that shape where to move. Whether it’s flipping a shape (reflection), spinning it (rotation), or stretching it (enlargement), we can represent these movements using a single \(2 \times 2\) matrix.

Quick Review: In this chapter, we focus on transformations where the Origin \( (0,0) \) stays in the same place. We call the starting shape the Object and the new shape the Image.

1. Moving Points with Matrices

To find out where a point goes after a transformation, we represent the point as a column vector and multiply it by the transformation matrix.

The Rule: Always put the matrix on the left and the point on the right.

\( \text{Matrix} \times \text{Object Point} = \text{Image Point} \)

For a matrix \( \mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) and a point \( (x, y) \):

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

Example: If the matrix is \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \) and our point is \( (3, 2) \):
\( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 3 \\ 2 \end{pmatrix} = \begin{pmatrix} (0 \times 3) + (1 \times 2) \\ (1 \times 3) + (0 \times 2) \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \end{pmatrix} \).
The point \( (3, 2) \) has moved to \( (2, 3) \)!

2. Finding the Matrix: The "Unit Vector" Trick

Don't worry if you forget a specific matrix! There is a "secret key" to finding any transformation matrix. We just look at what happens to two special points: \( I(1, 0) \) and \( J(0, 1) \).

  • The 1st column of your matrix is the new position of \( (1, 0) \).
  • The 2nd column of your matrix is the new position of \( (0, 1) \).

Did you know? These two points make up the Identity Matrix \( \mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \). This matrix is like the number "1" in multiplication—it doesn't change the shape at all!

3. Common Transformation Matrices

Here are the standard matrices you will encounter in the Pearson Edexcel Specification B exam. All of these use the origin \( (0, 0) \) as the center.

Reflections

  • Reflection in the x-axis (\( y = 0 \)): \( \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \)
  • Reflection in the y-axis (\( x = 0 \)): \( \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} \)
  • Reflection in the line \( y = x \): \( \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \)
  • Reflection in the line \( y = -x \): \( \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \)

Rotations (about the origin)

Remember: Positive angles are anticlockwise, negative angles are clockwise.

  • \( 90^\circ \) anticlockwise: \( \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \)
  • \( 180^\circ \) (either direction): \( \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \)
  • \( 270^\circ \) anticlockwise (or \( 90^\circ \) clockwise): \( \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \)

Enlargements (centre origin)

  • Enlargement with scale factor \( k \): \( \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \)

Key Takeaway: If a matrix is in the form \( \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \), it is always an enlargement by scale factor \( k \).

4. Combining Transformations

Sometimes, we perform one transformation and then follow it with another. For example, we might reflect a triangle and then rotate it. We can represent this combined transformation as a single matrix by multiplying the two individual matrices together.

CRITICAL RULE: The Order Matters!
If you do transformation \( \mathbf{B} \) first, followed by transformation \( \mathbf{A} \), the combined matrix is written as \( \mathbf{AB} \).

Think of it like reading from right to left: \( \mathbf{A} \leftarrow \mathbf{B} \leftarrow \text{Object} \). The matrix closest to the object is the one that happens first.

Step-by-Step Example:
1. Matrix \( \mathbf{P} \) is a reflection in the x-axis.
2. Matrix \( \mathbf{Q} \) is an enlargement by scale factor 3.
3. To find the single matrix for "Reflection \( \mathbf{P} \) followed by Enlargement \( \mathbf{Q} \)", calculate \( \mathbf{QP} \).
\( \mathbf{QP} = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 0 & -3 \end{pmatrix} \).

5. Using Determinants and Inverses

In previous chapters, you learned how to find the determinant (\( ad - bc \)) and the inverse of a matrix. These are very useful here!

  • The Determinant: The determinant of a transformation matrix tells you the Area Scale Factor. If the determinant is 5, the image's area is 5 times larger than the object's area. If it is negative, the orientation of the shape has been reversed (like in a reflection).
  • The Inverse: The inverse matrix \( \mathbf{M}^{-1} \) represents the reverse transformation. It takes the Image back to the Object.

Common Mistakes to Avoid

1. Mixing up columns and rows: When finding a matrix using \( (1, 0) \) and \( (0, 1) \), always write the new coordinates as columns, not rows.

2. Wrong multiplication order: Students often write \( \mathbf{BA} \) when the question says "\( \mathbf{B} \) followed by \( \mathbf{A} \)". Remember: The second transformation goes on the left!

3. Forgetting the origin: These \( 2 \times 2 \) matrices only work for rotations and enlargements centered at \( (0, 0) \). If the center is elsewhere, we use different vector methods (see the chapter on Transformations of the Plane).

Quick Review Summary:
- Point \( \times \) Matrix = Image (But write it as Matrix \( \times \) Point).
- Use \( I(1,0) \) and \( J(0,1) \) to build your matrix.
- \( \mathbf{AB} \) means \( \mathbf{B} \) then \( \mathbf{A} \).
- Determinant = Area scale factor.