Welcome to Matrices (CCEA AS 1: Pure Mathematics)
Welcome to one of the most powerful and exciting tools in Further Mathematics: Matrices! Whether you have encountered basic matrices before or are seeing them for the first time, do not worry if they feel unfamiliar. Think of a matrix simply as a tidy spreadsheet or rectangular grid of numbers that allows us to handle complex transformations, multi-variable equations, and geometric spaces all at once.
In this chapter for your Unit AS 1 exam, you will master the algebra of matrices, calculate determinants and inverses in both \(2 \times 2\) and \(3 \times 3\) dimensions, explore geometric transformations in the 2D plane, and uncover how systems of linear equations describe the geometry of intersecting planes.
1. Matrix Fundamentals & Basic Algebra
What is a Matrix?
A matrix is a rectangular array of numbers arranged in rows (horizontal) and columns (vertical). The size of a matrix is called its order or dimension, written as \(m \times n\), where \(m\) represents the number of rows and \(n\) represents the number of columns.
Memory Trick: Remember RC (like a Remote Control or Row then Column).
For example, the matrix \(A = \begin{pmatrix} 2 & 5 & -1 \\ 4 & 0 & 3 \end{pmatrix}\) has \(2\) rows and \(3\) columns, so its order is \(2 \times 3\).
Addition and Subtraction
You can add or subtract matrices if and only if they have the exact same dimensions. You simply add or subtract corresponding entries element by element.
For example: \(\begin{pmatrix} 1 & 3 \\ -2 & 4 \end{pmatrix} + \begin{pmatrix} 5 & -1 \\ 2 & 0 \end{pmatrix} = \begin{pmatrix} 1+5 & 3+(-1) \\ -2+2 & 4+0 \end{pmatrix} = \begin{pmatrix} 6 & 2 \\ 0 & 4 \end{pmatrix}\)
Scalar Multiplication
To multiply a matrix by a single number (a scalar \(k\)), multiply every single entry in the matrix by \(k\):
\(k \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} ka & kb \\ kc & kd \end{pmatrix}\)
Matrix Multiplication
Multiplying two matrices together is not done element by element. To compute the product \(AB\), the number of columns in \(A\) must equal the number of rows in \(B\).
If matrix \(A\) is of order \(m \times k\) and matrix \(B\) is of order \(k \times n\), their product \(AB\) is defined and has order \(m \times n\).
How to multiply: Take the row from the first matrix and the column from the second matrix, multiply matching pairs, and add the results together (the "row-into-column" rule):
\(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} ae + bg & af + bh \\ ce + dg & cf + dh \end{pmatrix}\)
Special Matrices & Key Algebraic Properties
• The Identity Matrix (\(I\)): Acts like the number \(1\) in ordinary multiplication. It is a square matrix with \(1\)s along the main diagonal and \(0\)s everywhere else.
\(I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\), \(I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) such that \(AI = IA = A\).
• The Zero Matrix (\(O\)): A matrix where every entry is \(0\).
• The Transpose (\(A^T\)): Swaps the rows and columns of matrix \(A\). If row 1 is \(\begin{pmatrix} 2 & 3 \end{pmatrix}\), it becomes column 1 in \(A^T\).
• Important Algebraic Rules:
1. Not Commutative: In general, \(AB \neq BA\). Order matters significantly!
2. Associative: \(A(BC) = (AB)C\)
3. Distributive: \(A(B + C) = AB + AC\)
4. Transpose Reversal Law: \((AB)^T = B^T A^T\) (Notice how the order reverses!)
Key Takeaway: Check dimensions before multiplying. If \(A\) is \(2 \times 3\) and \(B\) is \(3 \times 2\), \(AB\) exists and is \(2 \times 2\), while \(BA\) is \(3 \times 3\). Matrix multiplication is not commutative!
2. Determinants and Inverses
\(2 \times 2\) Determinants & Inverses
For a \(2 \times 2\) matrix \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\):
• Determinant: \(\det(A) = |A| = ad - bc\)
• Inverse: If \(\det(A) \neq 0\), the inverse \(A^{-1}\) is given by:
\(A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\)
Quick Rule: Swap the main diagonal elements (\(a\) and \(d\)), change the signs of the off-diagonal elements (\(b\) and \(c\)), and divide the whole matrix by the determinant.
By definition, multiplying a matrix by its inverse yields the identity matrix: \(A A^{-1} = A^{-1} A = I\).
Singular vs Non-Singular Matrices
• Singular Matrix: \(\det(A) = 0\). A singular matrix has no inverse (since dividing by zero is impossible).
• Non-Singular Matrix: \(\det(A) \neq 0\). A non-singular matrix always possesses a unique inverse.
\(3 \times 3\) Determinants
For a \(3 \times 3\) matrix \(M = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\), find the determinant by expanding across the top row (remembering the sign pattern \(+ \ - \ +\)):
\(\det(M) = a \begin{vmatrix} e & f \\ h & i \end{vmatrix} - b \begin{vmatrix} d & f \\ g & i \end{vmatrix} + c \begin{vmatrix} d & e \\ g & h \end{vmatrix}\)
\(\det(M) = a(ei - fh) - b(di - fg) + c(dh - eg)\)
\(3 \times 3\) Matrix Inverse (Step-by-Step Procedure)
Finding the inverse of a \(3 \times 3\) matrix is a structured 4-step process:
Step 1: Calculate \(\det(M)\). Ensure \(\det(M) \neq 0\). If it is \(0\), stop—the matrix is singular.
Step 2: Find the Matrix of Minors (\(M_{ij}\)). Replace each entry with the determinant of the \(2 \times 2\) matrix remaining when its row and column are crossed out.
Step 3: Form the Matrix of Cofactors (\(C\)). Apply the alternating sign grid to the minors:
\(\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}\)
So \(C_{ij} = (-1)^{i+j} M_{ij}\).
Step 4: Transpose the Cofactor Matrix to get the Adjugate, then Divide by \(\det(M)\):
\(\text{adj}(M) = C^T\)
\(M^{-1} = \frac{1}{\det(M)} \text{adj}(M) = \frac{1}{\det(M)} C^T\)
Key Determinant and Inverse Properties
• Product Determinant Rule: \(\det(AB) = \det(A)\det(B)\)
• Inverse Reversal Law: \((AB)^{-1} = B^{-1}A^{-1}\)
Key Takeaway: Never forget Step 4 when computing a \(3 \times 3\) inverse: you must transpose the cofactor matrix (\(C^T\)) before multiplying by \(\frac{1}{\det(M)}\).
3. 2D Geometric Linear Transformations
Matrices can transform geometric points in the \(xy\)-plane. A point \((x, y)\) is represented as a column vector \(\begin{pmatrix} x \\ y \end{pmatrix}\). When multiplied by a \(2 \times 2\) matrix \(M\), it produces the image point \((x', y')\):
\(\begin{pmatrix} x' \\ y' \end{pmatrix} = M \begin{pmatrix} x \\ y \end{pmatrix}\)
Did you know? You can find the transformation matrix for any linear operation simply by finding where the unit vectors \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) land. The image of \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) forms column 1, and the image of \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) forms column 2!
Standard \(2 \times 2\) Transformation Matrices
1. Rotation about the Origin:
Anticlockwise through angle \(\theta\): \(\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\)
Note: For clockwise rotation, substitute \(-\theta\), giving \(\begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}\).
2. Reflection:
Reflection in the line \(y = x\tan\theta\) (a line passing through the origin at angle \(\theta\) to the positive \(x\)-axis):
\(\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}\)
Common Special Cases:
• Reflection in the \(x\)-axis (\(\theta = 0^\circ\)): \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)
• Reflection in the \(y\)-axis (\(\theta = 90^\circ\)): \(\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\)
• Reflection in the line \(y = x\) (\(\theta = 45^\circ\)): \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)
• Reflection in the line \(y = -x\) (\(\theta = 135^\circ\)): \(\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\)
3. Enlargement (Scaling from Origin):
Scale factor \(k\), centre \((0,0)\): \(\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}\)
4. Two-Way Stretch:
Stretch with scale factor \(p\) parallel to the \(x\)-axis and scale factor \(q\) parallel to the \(y\)-axis: \(\begin{pmatrix} p & 0 \\ 0 & q \end{pmatrix}\)
5. Shears:
• Shear parallel to the \(x\)-axis: Points on the \(x\)-axis stay fixed; \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) moves to \(\begin{pmatrix} k \\ 1 \end{pmatrix}\): \(\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}\)
• Shear parallel to the \(y\)-axis: Points on the \(y\)-axis stay fixed; \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) moves to \(\begin{pmatrix} 1 \\ k \end{pmatrix}\): \(\begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix}\)
Composite Transformations
When applying transformation \(A\) followed by transformation \(B\), the combined single matrix is \(BA\).
Why? If \(\mathbf{x}\) is transformed first by \(A\), the new position is \(A\mathbf{x}\). Applying \(B\) to that result gives \(B(A\mathbf{x}) = (BA)\mathbf{x}\).
Determinant as Area Scale Factor
The determinant of a \(2 \times 2\) transformation matrix has a direct geometric meaning:
• Area Scale Factor: \(\text{New Area} = |\det(M)| \times \text{Original Area}\)
• Orientation: If \(\det(M) > 0\), the orientation of the shape is preserved (e.g., rotations, enlargements). If \(\det(M) < 0\), the shape has been flipped/reversed in orientation (e.g., reflections).
Key Takeaway: Always multiply transformation matrices in reverse chronological order: transformation \(A\) followed by \(B\) is calculated as \(BA\).
4. Invariant Points and Invariant Lines
Examiners love testing the difference between these two related but distinct ideas:
1. Invariant Points & Lines of Invariant Points
An invariant point is a point that does not move at all under the transformation:
\(M \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}\)
When you solve this equation, if you get a single solution like \((0,0)\), the origin is the only invariant point. If the equations reduce to a linear relationship like \(y = 2x\), then every point on that line stays exactly where it is. This is called a line of invariant points.
2. Invariant Lines
An invariant line is a line where points on the line map to points that remain on the same line. Individual points may slide along the line, but the line as an entire set remains unchanged.
Every line of invariant points is an invariant line, but NOT every invariant line is a line of invariant points!
How to Find Invariant Lines:
Case 1: Non-vertical lines (\(y = mx + c\))
1. Write a general point on the line: \(\begin{pmatrix} x \\ mx + c \end{pmatrix}\).
2. Transform the point: \(\begin{pmatrix} x' \\ y' \end{pmatrix} = M \begin{pmatrix} x \\ mx + c \end{pmatrix}\).
3. Substitute \(x'\) and \(y'\) into the equation of the line: \(y' = mx' + c\).
4. Equate coefficients of \(x\) and constant terms to solve for \(m\) and \(c\).
Case 2: Vertical lines (\(x = k\))
Always check vertical lines separately! Transform \(\begin{pmatrix} k \\ y \end{pmatrix}\) using \(M\) and check if the resulting \(x'\) coordinate is equal to \(k\) independently of \(y\).
Key Takeaway: For a line of invariant points, solve \(M\mathbf{r} = \mathbf{r}\). For an invariant line, transform \((x, mx+c)^T\) and require the image to satisfy \(y' = mx' + c\).
5. Systems of Linear Equations in 3 Variables
A system of three linear equations:
\(a_1 x + b_1 y + c_1 z = d_1\)
\(a_2 x + b_2 y + c_2 z = d_2\)
\(a_3 x + b_3 y + c_3 z = d_3\)
can be written cleanly in matrix form as \(A\mathbf{x} = \mathbf{b}\), where:
\(A = \begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{pmatrix}, \quad \mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} d_1 \\ d_2 \\ d_3 \end{pmatrix}\)
Geometric Interpretations of Solutions
Each linear equation in 3 variables represents a flat plane in 3D space.
1. Non-Singular Matrix (\(\det(A) \neq 0\)): Unique Solution
• The system has exactly one solution: \(\mathbf{x} = A^{-1}\mathbf{b}\).
• Geometric Meaning: The three planes intersect at a single unique point.
2. Singular Matrix (\(\det(A) = 0\)): Two Possibilities
When \(\det(A) = 0\), you use Gaussian elimination (row reduction) to inspect the system:
• Consistent (Infinitely Many Solutions):
Row reduction leads to an identity like \(0x + 0y + 0z = 0\). The equations do not contradict each other.
Geometric Meaning: The planes intersect along a common straight line (forming a sheaf of planes) or represent identical planes.
• Inconsistent (No Solutions):
Row reduction leads to a contradiction like \(0x + 0y + 0z = k\) (where \(k \neq 0\)).
Geometric Meaning: The planes do not share any common points. They may form a triangular prism (where pairs of planes intersect in parallel lines) or two/more planes are parallel and distinct.
Key Takeaway: \(\det(A) \neq 0\) means a single point of intersection. \(\det(A) = 0\) means either infinite solutions (sheaf/line) or zero solutions (prism/parallel planes).
6. Common Pitfalls & Examiner Tips
1. Invariant Points vs Invariant Lines: Do not confuse these! If an exam question asks for "invariant lines", setting \(M\mathbf{r} = \mathbf{r}\) only finds lines where points do not move. You must use \((x, mx+c)^T\) and solve for \(m\) and \(c\).
2. Transformation Multiplication Order: Remember that transformation \(A\) followed by \(B\) is written as \(BA\), not \(AB\).
3. Bracket vs Determinant Notation: Square or round brackets \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) mean a matrix (an array). Straight vertical bars \(\begin{vmatrix} a & b \\ c & d \end{vmatrix}\) mean a determinant (a single scalar value).
4. Adjugate Transpose Mistake: When computing \(3 \times 3\) inverses, always transpose the cofactor matrix. Forgetting this step is one of the most common mark-losing errors in AS 1.
5. Forgetting Vertical Lines: When finding invariant lines, always test the vertical line case \(x = k\) separately from \(y = mx + c\).
6. Reversal Laws: Remember that both transposes and inverses reverse order: \((AB)^{-1} = B^{-1}A^{-1}\) and \((AB)^T = B^T A^T\).