Welcome to Matrices: Unit 1 Pure Mathematics
Welcome to one of the most powerful and exciting topics in CCEA GCSE Further Mathematics: Matrices! Don't worry if matrices look unfamiliar at first—they are simply organised grids of numbers that give us a neat, systematic way to handle multiple equations and data sets at once.
Did you know? Matrices are used every day by computer graphics engines to render 3D video games, by search engines to rank web pages, and by engineers to model complex structures. In this chapter, we will master the foundations: from basic matrix arithmetic to finding determinants, inverses, and solving simultaneous equations using matrix methods.
Let's take this step-by-step to build your confidence and help you secure full marks in your Unit 1: Pure Mathematics (GFM11) exam!
---1. Matrix Basics and Notation
What is a Matrix?
A matrix (plural: matrices) is a rectangular array of numbers, variables, or expressions arranged in horizontal rows and vertical columns. We enclose matrices inside round brackets \(\begin{pmatrix} \dots \end{pmatrix}\).
Order (Dimensions) of a Matrix
The order or dimension of a matrix tells you its size and is always written as:
\(\text{Rows} \times \text{Columns}\) (\(m \times n\))
where \(m\) is the number of horizontal rows and \(n\) is the number of vertical columns.
Memory Trick: Remember RC (like a Remote Control or Row Column). Always count down the rows first, then across the columns.
• \(\begin{pmatrix} 3 & -1 \\ 2 & 5 \end{pmatrix}\) has \(2\) rows and \(2\) columns, so its order is \(2 \times 2\).
• \(\begin{pmatrix} 4 & 1 & 0 \\ 7 & -3 & 2 \end{pmatrix}\) has \(2\) rows and \(3\) columns, so its order is \(2 \times 3\).
• \(\begin{pmatrix} 5 \\ -2 \end{pmatrix}\) has \(2\) rows and \(1\) column, so its order is \(2 \times 1\) (often called a column vector).
Special Types of Matrices
There are a few special matrices you must know for your CCEA specification:
1. Square Matrix:
A matrix with the same number of rows and columns (e.g. \(2 \times 2\)).
2. Zero Matrix (or Null Matrix, \(\mathbf{0}\)):
A matrix where every single entry is zero. For a \(2 \times 2\) zero matrix:
\(\mathbf{0} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\)
3. Identity Matrix (\(\mathbf{I}\)):
A square matrix containing ones (\(1\)) along the main diagonal (from top-left to bottom-right) and zeros (\(0\)) everywhere else. For a \(2 \times 2\) matrix:
\(\mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)
Special Property of \(\mathbf{I}\): The identity matrix acts just like the number \(1\) in ordinary multiplication. For any compatible square matrix \(\mathbf{A}\):
\(\mathbf{AI} = \mathbf{IA} = \mathbf{A}\)
Key Takeaways for Section 1:
• Order is always \(\text{Rows} \times \text{Columns}\) (\(m \times n\)).
• \(\mathbf{I} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\) is the matrix equivalent of the number \(1\).
2. Matrix Arithmetic: Addition, Subtraction, and Scalar Multiplication
Addition and Subtraction
Matrix addition and subtraction are straightforward, but there is one strict rule:
Matrices MUST have the exact same order to be added or subtracted.
You simply add or subtract the corresponding elements (the numbers in the same positions):
\(\begin{pmatrix} a & b \\ c & d \end{pmatrix} + \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} a+e & b+f \\ c+g & d+h \end{pmatrix}\)
\(\begin{pmatrix} a & b \\ c & d \end{pmatrix} - \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} a-e & b-f \\ c-g & d-h \end{pmatrix}\)
Example:
If \(\mathbf{A} = \begin{pmatrix} 4 & -2 \\ 1 & 3 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 5 & 1 \\ -4 & 6 \end{pmatrix}\), then:
\(\mathbf{A} + \mathbf{B} = \begin{pmatrix} 4+5 & -2+1 \\ 1+(-4) & 3+6 \end{pmatrix} = \begin{pmatrix} 9 & -1 \\ -3 & 9 \end{pmatrix}\)
\(\mathbf{A} - \mathbf{B} = \begin{pmatrix} 4-5 & -2-1 \\ 1-(-4) & 3-6 \end{pmatrix} = \begin{pmatrix} -1 & -3 \\ 5 & -3 \end{pmatrix}\)
Scalar Multiplication
A scalar is just a regular ordinary number (like \(3\), \(-2\), or \(\frac{1}{2}\)). To multiply a matrix by a scalar \(k\), multiply every single entry inside the matrix by \(k\):
\(k \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} ka & kb \\ kc & kd \end{pmatrix}\)
Example:
If \(\mathbf{A} = \begin{pmatrix} 3 & -4 \\ 2 & 0 \end{pmatrix}\), find \(3\mathbf{A}\):
\(3\mathbf{A} = 3 \begin{pmatrix} 3 & -4 \\ 2 & 0 \end{pmatrix} = \begin{pmatrix} 3(3) & 3(-4) \\ 3(2) & 3(0) \end{pmatrix} = \begin{pmatrix} 9 & -12 \\ 6 & 0 \end{pmatrix}\)
Key Takeaways for Section 2:
• Only add or subtract matrices if they have the exact same dimensions.
• When multiplying by a scalar, distribute it to every element inside the matrix.
3. Matrix Multiplication
Multiplying two matrices together is different from multiplying ordinary numbers. It requires a specific row-by-column procedure.
Can We Multiply? (Compatibility Rule)
To multiply matrix \(\mathbf{A}\) by matrix \(\mathbf{B}\) (written as \(\mathbf{AB}\)):
The number of columns in \(\mathbf{A}\) must equal the number of rows in \(\mathbf{B}\).
If \(\mathbf{A}\) has order \((\mathbf{m} \times \mathbf{k})\) and \(\mathbf{B}\) has order \((\mathbf{k} \times \mathbf{n})\):
• The inner numbers match (\(k = k\)), so multiplication is possible.
• The outer numbers give the resulting order: \((m \times n)\).
The Multiplication Process: "Row into Column"
To find each entry in the product matrix, move across the row of the first matrix and down the column of the second matrix, multiplying paired entries and adding the results.
Formula for \(2 \times 2\) by \(2 \times 2\):
\(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{pmatrix}\)
Formula for \(2 \times 2\) by \(2 \times 1\):
\(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}\)
Step-by-Step Worked Example
Calculate \(\mathbf{AB}\) where \(\mathbf{A} = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 5 & -1 \\ 0 & 2 \end{pmatrix}\):
• Top-left entry: \((\text{Row } 1 \text{ of } \mathbf{A}) \times (\text{Col } 1 \text{ of } \mathbf{B}) = (2 \times 5) + (3 \times 0) = 10 + 0 = 10\)
• Top-right entry: \((\text{Row } 1 \text{ of } \mathbf{A}) \times (\text{Col } 2 \text{ of } \mathbf{B}) = (2 \times -1) + (3 \times 2) = -2 + 6 = 4\)
• Bottom-left entry: \((\text{Row } 2 \text{ of } \mathbf{A}) \times (\text{Col } 1 \text{ of } \mathbf{B}) = (1 \times 5) + (4 \times 0) = 5 + 0 = 5\)
• Bottom-right entry: \((\text{Row } 2 \text{ of } \mathbf{A}) \times (\text{Col } 2 \text{ of } \mathbf{B}) = (1 \times -1) + (4 \times 2) = -1 + 8 = 7\)
Result:
\(\mathbf{AB} = \begin{pmatrix} 10 & 4 \\ 5 & 7 \end{pmatrix}\)
Crucial Fact: Matrix Multiplication is NOT Commutative
In standard arithmetic, \(3 \times 5 = 5 \times 3\). But in matrix algebra:
\(\mathbf{AB} \neq \mathbf{BA}\) in general!
The order in which you write the matrices matters completely. Changing the order usually gives a completely different answer, or makes multiplication impossible.
Key Takeaways for Section 3:
• Check dimensions: \((m \times k) \times (k \times n) \to (m \times n)\).
• Multiply Row \(\times\) Column: move across the row, down the column, multiply pairs, and add.
• Order matters: \(\mathbf{AB} \neq \mathbf{BA}\).
4. Determinants and Inverses of \(2 \times 2\) Matrices
The Determinant of a \(2 \times 2\) Matrix
The determinant is a single number calculated from a square matrix. For a matrix \(\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), the determinant is denoted as \(\det(\mathbf{A})\) or \(|\mathbf{A}|\) and is defined as:
\(\det(\mathbf{A}) = |\mathbf{A}| = ad - bc\)
Worked Example:
Find the determinant of \(\mathbf{M} = \begin{pmatrix} 3 & -2 \\ 4 & 5 \end{pmatrix}\):
\(\det(\mathbf{M}) = (3)(5) - (-2)(4) = 15 - (-8) = 15 + 8 = 23\)
Singular vs Non-Singular Matrices
• If \(\det(\mathbf{A}) = 0\), \(\mathbf{A}\) is called a singular matrix. A singular matrix does not have an inverse.
• If \(\det(\mathbf{A}) \neq 0\), \(\mathbf{A}\) is called a non-singular matrix. A non-singular matrix has a unique inverse.
The Inverse Matrix (\(\mathbf{A}^{-1}\))
The inverse of matrix \(\mathbf{A}\), written as \(\mathbf{A}^{-1}\), is the matrix that "undoes" \(\mathbf{A}\). When multiplied together in either order, they give the identity matrix \(\mathbf{I}\):
\(\mathbf{A}\mathbf{A}^{-1} = \mathbf{A}^{-1}\mathbf{A} = \mathbf{I}\)
Formula for the Inverse of a \(2 \times 2\) Matrix
For \(\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), where \(\det(\mathbf{A}) = ad - bc \neq 0\):
\(\mathbf{A}^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\)
Two-Step Memory Rule:
1. Swap the elements on the leading diagonal (\(a\) and \(d\) swap places).
2. Change the signs of the other two elements (\(b\) becomes \(-b\), and \(c\) becomes \(-c\)).
3. Multiply the entire resulting matrix by \(\frac{1}{\det(\mathbf{A})}\).
Worked Example: Finding an Inverse
Find the inverse of \(\mathbf{A} = \begin{pmatrix} 4 & 2 \\ 3 & 2 \end{pmatrix}\):
Step 1: Calculate the determinant:
\(\det(\mathbf{A}) = ad - bc = (4)(2) - (2)(3) = 8 - 6 = 2\)
Step 2: Rearrange the matrix elements (swap main diagonal, negate off-diagonal):
\(\begin{pmatrix} 2 & -2 \\ -3 & 4 \end{pmatrix}\)
Step 3: Multiply by \(\frac{1}{\det(\mathbf{A})}\):
\(\mathbf{A}^{-1} = \frac{1}{2} \begin{pmatrix} 2 & -2 \\ -3 & 4 \end{pmatrix} = \begin{pmatrix} 1 & -1 \\ -1.5 & 2 \end{pmatrix}\)
Key Takeaways for Section 4:
• Determinant: \(\det(\mathbf{A}) = ad - bc\).
• If \(\det(\mathbf{A}) = 0\), the matrix is singular and has no inverse.
• Inverse formula: \(\mathbf{A}^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\).
5. Solving Matrix Equations & Systems of Simultaneous Equations
Solving Matrix Equations (Pre-Multiplication vs Post-Multiplication)
Because matrix multiplication is not commutative, there is no "division" in matrix algebra. Instead, we multiply by the inverse matrix. The side on which you multiply is critical!
Case 1: \(\mathbf{A}\) is on the left (\(\mathbf{AX} = \mathbf{B}\))
Pre-multiply both sides by \(\mathbf{A}^{-1}\):
\(\mathbf{A}^{-1}\mathbf{AX} = \mathbf{A}^{-1}\mathbf{B}\)
\(\mathbf{IX} = \mathbf{A}^{-1}\mathbf{B}\)
\(\mathbf{X} = \mathbf{A}^{-1}\mathbf{B}\)
Case 2: \(\mathbf{A}\) is on the right (\(\mathbf{XA} = \mathbf{B}\))
Post-multiply both sides by \(\mathbf{A}^{-1}\):
\(\mathbf{XA}\mathbf{A}^{-1} = \mathbf{B}\mathbf{A}^{-1}\)
\(\mathbf{XI} = \mathbf{B}\mathbf{A}^{-1}\)
\(\mathbf{X} = \mathbf{B}\mathbf{A}^{-1}\)
Solving Simultaneous Linear Equations using Matrices
A system of two linear simultaneous equations in two unknowns \(x\) and \(y\):
\(ax + by = p\)
\(cx + dy = q\)
can be converted directly into the matrix form \(\mathbf{AX} = \mathbf{B}\):
\(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} p \\ q \end{pmatrix}\)
To solve for \(\begin{pmatrix} x \\ y \end{pmatrix}\), pre-multiply by the inverse of the coefficient matrix:
\(\begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \begin{pmatrix} p \\ q \end{pmatrix}\)
Full Exam-Style Walkthrough
Solve the following simultaneous equations using a matrix method:
\(3x + 2y = 16\)
\(4x - y = 7\)
Step 1: Write in matrix form \(\mathbf{AX} = \mathbf{B}\)
\(\begin{pmatrix} 3 & 2 \\ 4 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 16 \\ 7 \end{pmatrix}\)
Step 2: Find the determinant of the coefficient matrix
\(\det(\mathbf{A}) = ad - bc = (3)(-1) - (2)(4) = -3 - 8 = -11\)
Step 3: Find the inverse matrix \(\mathbf{A}^{-1}\)
\(\mathbf{A}^{-1} = \frac{1}{-11} \begin{pmatrix} -1 & -2 \\ -4 & 3 \end{pmatrix} = -\frac{1}{11} \begin{pmatrix} -1 & -2 \\ -4 & 3 \end{pmatrix}\)
Step 4: Pre-multiply \(\mathbf{B}\) by \(\mathbf{A}^{-1}\)
\(\begin{pmatrix} x \\ y \end{pmatrix} = -\frac{1}{11} \begin{pmatrix} -1 & -2 \\ -4 & 3 \end{pmatrix} \begin{pmatrix} 16 \\ 7 \end{pmatrix}\)
Multiply the \(2 \times 2\) matrix and \(2 \times 1\) column vector first:
\(\begin{pmatrix} (-1)(16) + (-2)(7) \\ (-4)(16) + (3)(7) \end{pmatrix} = \begin{pmatrix} -16 - 14 \\ -64 + 21 \end{pmatrix} = \begin{pmatrix} -30 \\ -43 \end{pmatrix}\)
Now apply the scalar \(-\frac{1}{11}\):
\(\begin{pmatrix} x \\ y \end{pmatrix} = -\frac{1}{11} \begin{pmatrix} -30 \\ -43 \end{pmatrix} = \begin{pmatrix} \frac{30}{11} \\ \frac{43}{11} \end{pmatrix}\)
Step 5: Explicitly state the values of \(x\) and \(y\)
\(x = \frac{30}{11}\), \(y = \frac{43}{11}\)
Key Takeaways for Section 5:
• For \(\mathbf{AX} = \mathbf{B}\), always calculate \(\mathbf{X} = \mathbf{A}^{-1}\mathbf{B}\) (pre-multiply).
• Keep the fraction \(\frac{1}{\det(\mathbf{A})}\) at the front until after you multiply the matrices—it keeps calculations much simpler!
• Always write out your final answer as \(x = \dots\) and \(y = \dots\).
6. Examiner Pitfalls and Common Mistakes
Make sure you avoid these common traps reported by CCEA examiners:
1. Wrong Multiplication Order when Solving Equations:
Writing \(\mathbf{X} = \mathbf{B}\mathbf{A}^{-1}\) instead of \(\mathbf{X} = \mathbf{A}^{-1}\mathbf{B}\) when solving \(\mathbf{AX} = \mathbf{B}\). Remember: since \(\mathbf{A}\) is on the left of \(\mathbf{X}\), \(\mathbf{A}^{-1}\) must be applied to the left of \(\mathbf{B}\).
2. Determinant Sign Errors:
Forgetting double negatives in \(ad - bc\). For example, if \(b = -3\) and \(c = 2\), then \(-bc = -(-3 \times 2) = +6\). Always write out the brackets carefully.
3. Inverse Matrix Construction Errors:
Mixing up the swaps and sign changes when finding \(\mathbf{A}^{-1}\).
• Swap the leading diagonal: \(a \leftrightarrow d\).
• Change signs on the other diagonal: \(b \to -b\), \(c \to -c\) (do not swap their positions!).
4. Scalar Multiplication Misses:
When multiplying a matrix by a scalar \(k\), make sure you multiply every single entry in all rows and columns, not just the top row.
5. Leaving Answers Trapped in Matrix Form:
If an exam question asks you to "find the values of \(x\) and \(y\)", do not stop at \(\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ -5 \end{pmatrix}\). Clearly write your final conclusion: \(x = 2\), \(y = -5\).
Quick Revision Checklist
Before sitting your Unit 1 examination, make sure you can confidently:
• State the order of any matrix as \(\text{rows} \times \text{columns}\).
• Add, subtract, and multiply matrices by a scalar.
• Multiply compatible matrices using row-by-column multiplication.
• Calculate the determinant of a \(2 \times 2\) matrix (\(ad - bc\)).
• Identify singular matrices (\(\det = 0\)).
• Find the inverse of a non-singular \(2 \times 2\) matrix.
• Solve matrix equations and systems of simultaneous equations using the matrix inverse method.