Further Mathematics (9665) | Matrices and transformations

👋 Welcome to Matrices and Transformations!

Hello! If you've reached this chapter, you’re diving into one of the most exciting areas of Further Mathematics. Matrices are more than just grids of numbers; they are the fundamental tool used to describe movement, resizing, and distortion in geometry and computer graphics.

In this chapter, we will learn how to calculate with matrices (the algebra part) and, crucially, how to interpret what these calculations mean visually (the geometry part). Don't worry if this seems tricky at first—we'll break down the concepts step-by-step!

Chapter Focus: FPP1.1 Matrices and Transformations

Our focus in AS Further Mathematics (FPP1) is primarily on \(2 \times 2\) matrices for geometric transformations in the \(x-y\) plane, but we must also be comfortable with the algebra of \(3 \times 3\) matrices.

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1. Matrix Algebra Fundamentals (Up to \(3 \times 3\))

A matrix is simply a rectangular array of numbers. Its size (or order) is defined by its number of rows (\(m\)) and columns (\(n\)), written as \(m \times n\).

Matrix Addition and Subtraction

This is the easiest operation! You can only add or subtract matrices if they have the exact same order. You simply add or subtract corresponding elements.

Example:
$$\begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix} + \begin{pmatrix} 1 & 5 \\ 4 & -2 \end{pmatrix} = \begin{pmatrix} 2+1 & 1+5 \\ 0+4 & 3-2 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 4 & 1 \end{pmatrix}$$

Matrix Multiplication (\(A \times B\))

This is where students often make mistakes. Matrix multiplication is not commutative (usually \(AB \neq BA\)), and it follows a specific "row by column" rule.

Compatibility Rule:

You can only multiply matrix \(A\) (size \(m \times n\)) by matrix \(B\) (size \(p \times q\)) if the number of columns in \(A\) equals the number of rows in \(B\). That is, \(n\) must equal \(p\).

The resulting matrix \(AB\) will have the size \(m \times q\).

Step-by-Step Multiplication:

To find the element in the \(i\)-th row and \(j\)-th column of \(AB\), you multiply the elements of the \(i\)-th row of \(A\) by the corresponding elements of the \(j\)-th column of \(B\) and sum the products (a dot product).

Analogy: Imagine a boat (Row A) traveling down a river (Column B). Every time they cross paths, they multiply their values, and then you add the results!

The Identity Matrix (\(I\))

The Identity Matrix acts like the number '1' in standard multiplication. When you multiply any matrix \(A\) by the identity matrix \(I\) (of the appropriate size), you get \(A\) back again: \(AI = IA = A\).

It is a square matrix with 1s on the main diagonal and 0s everywhere else.

• \(2 \times 2\) Identity: $$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$
• \(3 \times 3\) Identity (We need to know this one too!): $$I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$

The Transpose of a Matrix (\(A^{\text{T}}\))

The transpose is found by simply switching the rows and columns.

Example: If $$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$$, then $$A^{\text{T}} = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$$

Key Transpose Result (Syllabus Requirement):

The transpose of a product is the product of the transposes in reverse order: $$(\mathbf{AB})^{\text{T}} = \mathbf{B}^{\text{T}}\mathbf{A}^{\text{T}}$$

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2. Determinants and The Inverse of a \(2 \times 2\) Matrix

The Determinant of a \(2 \times 2\) Matrix

The determinant, denoted \(\det(A)\) or \(|A|\), is a single scalar value calculated from a square matrix. For transformations, it tells us the area scale factor (more on this later!).

For a \(2 \times 2\) matrix $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$, the determinant is: $$ \det(A) = ad - bc $$

The Inverse of a \(2 \times 2\) Matrix (\(A^{-1}\))

The inverse of a matrix \(A\) is a matrix \(A^{-1}\) such that when you multiply them together, you get the Identity Matrix: \(AA^{-1} = A^{-1}A = I\).

Step-by-Step Inverse Calculation:

If $$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$, then the inverse is: $$ A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} $$

Trick: Swap \(a\) and \(d\), and change the signs of \(b\) and \(c\). Then divide the whole thing by the determinant!

Singular and Non-Singular Matrices

The determinant acts as a crucial "gatekeeper" for the inverse:

• If \(\det(A) \neq 0\), the matrix is non-singular. An inverse \(A^{-1}\) exists.
• If \(\det(A) = 0\), the matrix is singular. The term \(\frac{1}{0}\) is undefined, so the matrix has no inverse. Geometrically, this means the transformation squashes all area down to zero (e.g., mapping the entire plane onto a single line).

Key Inverse Result (Syllabus Requirement):

The inverse of a product is the product of the inverses in reverse order: $$(\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}$$

Did you know? This reverse order rule, seen in both transpose and inverse, is sometimes called the "socks and shoes" rule. To undo putting on socks then shoes, you must take off your shoes first, then your socks!

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3. Matrices as Geometric Transformations (2D)

A \(2 \times 2\) matrix $M$ transforms a point \((x, y)\) in the plane to a new point \((x', y')\) using the relationship: $$ \begin{pmatrix} x' \\ y' \end{pmatrix} = M \begin{pmatrix} x \\ y \end{pmatrix} $$

Finding the Transformation Matrix (The Unit Square Trick)

If you forget the standard matrix for a transformation, remember that the matrix columns are simply the coordinates of where the unit vectors $\mathbf{i} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\mathbf{j} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ are mapped to.

If $\mathbf{i} \to \begin{pmatrix} a \\ c \end{pmatrix}$ and $\mathbf{j} \to \begin{pmatrix} b \\ d \end{pmatrix}$, then the matrix is $$ M = \begin{pmatrix} a & b \\ c & d \end{pmatrix} $$.

Standard Transformations (About the Origin)

1. Enlargement (Centre Origin)

If the scale factor is \(k\): $$ E = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} $$

2. Stretches (Parallel to Axes)

• Stretch parallel to the \(x\)-axis (or perpendicular to the \(y\)-axis) with factor \(k\): $$ S_x = \begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix} $$
• Stretch parallel to the \(y\)-axis (or perpendicular to the \(x\)-axis) with factor \(k\): $$ S_y = \begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix} $$

3. Rotation (About the Origin)

Rotation through an angle \(\theta\) anti-clockwise about the origin: $$ R = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} $$

Memory Aid: The first column, $\begin{pmatrix} \cos \theta \\ \sin \theta \end{pmatrix}$, is always the point where (1, 0) is mapped to.

4. Reflections (In lines through the Origin)

These standard matrices are usually given in the formulae booklet, but it helps to know them:

• 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 \(y=-x\): $$ \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} $$

5. Shears

A shear shifts points parallel to a fixed line, usually the \(x\)-axis or \(y\)-axis.

• Shear parallel to the \(x\)-axis (shear factor \(k\)): $$ \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix} $$ (Points on the \(x\)-axis are invariant.)
• Shear parallel to the \(y\)-axis (shear factor \(k\)): $$ \begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix} $$ (Points on the \(y\)-axis are invariant.)

Note: If a matrix represents a shear and the line of invariant points is NOT the x or y axis, the question will inform you it is a shear.

Combinations of Transformations

To find the matrix for a combination of transformations, you multiply the individual matrices.

CRUCIAL RULE: Order Matters!

If transformation \(T_1\) is performed first, followed by transformation \(T_2\), the combined matrix \(M\) is calculated as: $$ M = T_2 T_1 $$

Think of it this way: the matrix closest to the coordinate vector $\begin{pmatrix} x \\ y \end{pmatrix}$ acts first. You read the operations from right to left.

Example: A point is rotated 90° clockwise ($T_1$) then reflected in $y=x$ ($T_2$). The combined matrix is \(M = T_2 T_1\).

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4. Geometric Properties and Invariants

The Determinant and Area

The most important geometric interpretation of the \(2 \times 2\) determinant is its link to area:

• The magnitude of the determinant, \(|\det(A)|\), is the Area Scale Factor of the transformation.
• If a shape has an area \(A_0\), the transformed area \(A'\) is: $$ A' = |\det(A)| \times A_0 $$

Interpreting a Negative Determinant

If \(\det(A)\) is negative, it means the transformation includes a reflection (an orientation reversal).

Example: A rotation (no reflection) has a determinant of +1. A reflection has a determinant of -1. A rotation followed by a reflection will have a determinant of -1.

Invariant Points

An invariant point is a point that does not move under the transformation. If the transformation is represented by matrix \(A\), then the point $\mathbf{x} = \begin{pmatrix} x \\ y \end{pmatrix}$ is invariant if: $$ \mathbf{A}\mathbf{x} = \mathbf{x} $$

Since \(\mathbf{x} = I\mathbf{x}\) (where \(I\) is the identity matrix), we can rearrange this equation to solve for \(\mathbf{x}\): $$ A\mathbf{x} - I\mathbf{x} = \mathbf{0} $$ $$ (A - I)\mathbf{x} = \mathbf{0} $$

This is a system of linear equations that you can solve simultaneously to find the coordinates of the invariant points (or lines of invariant points).

Invariant Lines

An invariant line is a line where every point on the line is mapped to a point *on the same line*.

Note the difference:

• Line of Invariant Points: Every point on the line stays exactly where it is. (This happens for reflections and shears).
• Invariant Line: The line itself is fixed, but the individual points on it might move along the line to new positions. (This happens for stretches).

How to Find Invariant Lines:

We usually look for lines passing through the origin, \(y = mx\). We assume a general point $(x, mx)$ is mapped to a new point $(x', y')$, and that $(x', y')$ must also satisfy the line equation, so $y' = m x'$.

1. Set up the transformation: $$ \begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ mx \end{pmatrix} $$ 2. Calculate $x'$ and $y'$ in terms of $x$ and $m$. 3. Substitute these into $y' = m x'$. 4. Simplify and solve for \(m\). (You will often find a quadratic for \(m\).)

If you find that the entire line consists of invariant points, the transformation must satisfy \(y = mx\) for all points, and it will be one of the special cases (reflection or shear).

Keep practicing your algebra, especially matrix multiplication, as mistakes there are the most common cause of error in transformation questions. Good luck!