Welcome to the Matrix Algebra Study Guide!
Hello! If you've reached this chapter, you are now diving into one of the most powerful and exciting areas of Further Mathematics. Don't worry if matrices feel strange at first—they are simply a neat, organized way to handle lots of numbers and equations simultaneously.
Matrices allow us to perform complex transformations (like rotating shapes in 2D and 3D space) and solve large systems of linear equations with efficiency. By the end of this chapter, you'll be a master of matrix operations, determinants, inverses, and the crucial concepts of eigenvectors and eigenvalues.
What is a Matrix?
Think of a matrix as a rectangular array (a fancy spreadsheet!) of numbers arranged in rows and columns.
- The order of a matrix is defined by its number of rows and then its number of columns, written as \(m \times n\).
Section 1: Fundamentals of Matrix Algebra (FPP1.1 Review)
1.1 Basic Operations
Addition and Subtraction
You can only add or subtract matrices if they have the exact same order. You simply add or subtract corresponding elements.
Example:
If \(A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) and \(B = \begin{pmatrix} 5 & 0 \\ -1 & 2 \end{pmatrix}\), then \(A+B = \begin{pmatrix} 1+5 & 2+0 \\ 3+(-1) & 4+2 \end{pmatrix} = \begin{pmatrix} 6 & 2 \\ 2 & 6 \end{pmatrix}\).
Scalar Multiplication
A scalar is just a normal number. To multiply a matrix by a scalar, you multiply every element in the matrix by that number.
Matrix Multiplication (Up to \(3 \times 3\))
This is often the first major challenge, but with practice, it becomes straightforward!
Rule Check: To multiply matrix \(A\) (order \(m \times n\)) by matrix \(B\) (order \(p \times q\)), the number of columns in \(A\) must equal the number of rows in \(B\). That is, \(n = p\).
The resulting matrix \(AB\) will have the order \(m \times q\).
Process: "Row by Column"
The element in the $i$-th row and $j$-th column of the product \(AB\) is found by multiplying the elements of the $i$-th row of \(A\) by the corresponding elements of the $j$-th column of \(B\), and then summing the results.
Analogy: Imagine you are sliding the rows of the first matrix over the columns of the second. Each pair you land on gets multiplied, and the results are added up.
Important Concept: Non-Commutativity
Unlike regular numbers where \(xy = yx\), for matrices, matrix multiplication is generally NOT commutative.
\(\mathbf{AB \neq BA}\) (even if both products exist).
Common Mistake: Assuming \(AB\) and \(BA\) are the same. Always check the required order!
1.2 Identity and Transpose Matrices
The Identity Matrix, \(I\)
The Identity Matrix, denoted \(I\), is the matrix equivalent of the number 1. When you multiply any matrix \(A\) by the identity matrix \(I\), you get \(A\) back.
\(AI = IA = A\)
It is a square matrix (rows = columns) with 1s on the leading diagonal (top-left to bottom-right) and 0s everywhere else.
\(2 \times 2\) Identity: \(I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)
\(3 \times 3\) Identity: \(I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\)
The Transpose Matrix, \(A^T\)
The Transpose of a matrix \(A\), denoted \(A^T\), is found by swapping the rows and the columns.
Example: If \(A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}\), then \(A^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}\).
Key Transpose Result (Syllabus Requirement):
The transpose of a product reverses the order:
\((AB)^T = B^T A^T\)
Quick Takeaway for Section 1: Master R-by-C multiplication and remember that matrices usually don't commute. The identity matrix is your friend, and the transpose reverses multiplication order.
Section 2: Determinants and Inverses
The determinant is a single number calculated from a square matrix. It tells us critical information about the matrix, especially whether it has an inverse.
2.1 Determinant of a \(2 \times 2\) Matrix
For a matrix \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), the determinant, \(\det(A)\) or \(|A|\), is:
$$\det(A) = ad - bc$$
Memory Aid: Multiply the elements on the main diagonal (\(a\) and \(d\)) and subtract the product of the elements on the other diagonal (\(b\) and \(c\)).
2.2 The Inverse of a \(2 \times 2\) Matrix
The Inverse Matrix, \(A^{-1}\), is the matrix such that \(A A^{-1} = A^{-1} A = I\).
The formula for the inverse of \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is:
$$A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$
Step-by-Step for \(2 \times 2\) Inverse:
- Calculate the determinant, \(\det(A) = ad - bc\).
- Swap \(a\) and \(d\).
- Change the signs of \(b\) and \(c\).
- Multiply the resulting matrix by \(\frac{1}{\det(A)}\).
Singular and Non-Singular Matrices
A matrix \(A\) is non-singular if \(\det(A) \neq 0\). If it is non-singular, its inverse \(A^{-1}\) exists.
A matrix \(A\) is singular if \(\det(A) = 0\). If it is singular, the fraction \(\frac{1}{0}\) is undefined, meaning the inverse does not exist.
2.3 Determinant of a \(3 \times 3\) Matrix (FP2 Level)
Calculating the determinant of a \(3 \times 3\) matrix requires the method of cofactor expansion.
For \(A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}\), you can expand along any row or column. Usually, the first row is used:
Step-by-Step: Cofactor Expansion
- Choose a row or column (choose one with zeros if possible, it makes life easier!).
- Use the alternating sign pattern (known as the cofactor signs):
\(\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}\) - For the first element, multiply the element by the determinant of the \(2 \times 2\) matrix remaining when that element's row and column are blocked out (this is the minor).
- Repeat for the second and third elements, remembering to apply the correct sign from the pattern.
- Sum the results.
2.4 The Inverse of a \(3 \times 3\) Matrix (FP2 Level)
Finding the inverse of a \(3 \times 3\) matrix is lengthy, but the process follows the definition:
$$A^{-1} = \frac{1}{\det(A)} \text{Adj}(A)$$
Where \(\text{Adj}(A)\) is the adjugate matrix (the transpose of the matrix of cofactors).
Encouragement: While the steps are many (finding 9 minors, applying signs to get 9 cofactors, transposing them, then dividing by the determinant), if you break it down, it's just repeating the \(2 \times 2\) determinant calculation many times. Be neat and organized!
2.5 Determinant of a Product
Key Result (Syllabus Requirement):
The determinant of a product of matrices is the product of their determinants. This holds for both \(2 \times 2\) and \(3 \times 3\) matrices:
$$\det(AB) = \det(A) \det(B)$$
Key Inverse Result (Syllabus Requirement):
The inverse of a product reverses the order:
$$(AB)^{-1} = B^{-1} A^{-1}$$
Quick Takeaway for Section 2: Determinants tell you if an inverse exists (if \(\det \neq 0\)). The methods for \(2 \times 2\) and \(3 \times 3\) are different, with \(3 \times 3\) relying on cofactors.
Section 3: Matrices and Geometric Transformations
Matrices are powerful tools for transforming points and shapes in geometry. If a point is represented by a column vector \(\mathbf{x}\), the transformed point \(\mathbf{x}'\) is given by \(\mathbf{x}' = M \mathbf{x}\), where \(M\) is the transformation matrix.
3.1 Transformations in 2D (\(2 \times 2\) Matrices)
The syllabus covers standard transformations that have the origin as the centre of transformation. You should be familiar with the matrices for:
- Rotation (about the origin by angle \(\theta\)): $$R = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$
- Reflection (in a line through the origin, e.g., the x-axis, y-axis, or \(y=x\)).
- Stretch (parallel to the x-axis or y-axis).
- Enlargement (centre at the origin, scale factor \(k\)): $$\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$$
-
Shear (e.g., parallel to the x-axis):
$$\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$$
Note: You only need to be able to recognise the matrix for a shear parallel to the x or y-axis. If the invariant line is not one of the axes, the question will inform you it is a shear.
Combinations of Transformations
If a transformation \(T_1\) (matrix \(M_1\)) is followed by a transformation \(T_2\) (matrix \(M_2\)), the combined transformation \(T\) is represented by the matrix \(M = M_2 M_1\).
Order Matters! Just like function composition, the matrix applied *first* goes on the right, next to the vector.
Determinant and Area Scale Factor (2D)
The absolute value of the determinant, \(|\det(M)|\), gives the area scale factor of the transformation.
If \(\det(M)\) is negative, it means the transformation includes a reflection (it reverses the orientation).
3.2 Transformations in 3D (\(3 \times 3\) Matrices, FP2 Level)
In three dimensions, matrices transform vectors \(\begin{pmatrix} x \\ y \\ z \end{pmatrix}\).
Rotations
You only need to know rotations about the coordinate axes (x, y, or z).
Did you know? These matrices are given in the formulae booklet, but understanding how they work is helpful. A rotation about the x-axis leaves \(x\) unchanged, so the first column/row looks like \(\begin{pmatrix} 1 & 0 & 0 \end{pmatrix}\).
Reflections
You must be able to work with reflections in the following specific planes (which all pass through the origin):
\(x=0\) (the \(yz\)-plane), \(y=0\), \(z=0\).
And the planes where coordinates are equal: \(x=y\), \(x=z\), \(y=z\).
Example: Reflection in the \(xy\)-plane (\(z=0\)). This flips the sign of \(z\) but keeps \(x\) and \(y\) the same: $$M = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}$$
Determinant and Volume Scale Factor (3D)
For a \(3 \times 3\) matrix \(M\), the absolute value of the determinant, \(|\det(M)|\), gives the volume scale factor of the transformation.
3.3 Invariant Points and Invariant Lines
An Invariant Point is a point whose position does not change after the transformation \(M\).
If \(\mathbf{x}\) is an invariant point, then:
$$M\mathbf{x} = \mathbf{x}$$
This can be rewritten as:
$$M\mathbf{x} = I\mathbf{x}$$
$$(M - I)\mathbf{x} = \mathbf{0}$$
An Invariant Line is a line where every point on the line is mapped back onto the same line (the points on the line might move, but the line itself stays in the same place).
Struggling? Think of a mirror. The mirror itself is an invariant plane. If you stand exactly on the mirror, you stay on the mirror (invariant line/plane). If you stand on the line of the hinge, that is an invariant point.
Quick Takeaway for Section 3: Matrices move points! The determinant is the scale factor for area (2D) or volume (3D). Remember the combination order \(M_2 M_1\). Invariant points satisfy \((M-I)\mathbf{x} = \mathbf{0}\).
Section 4: Eigenvalues and Eigenvectors (FP2 Level)
These are the most abstract concepts in the matrix algebra chapter, but they are also incredibly useful in advanced applications like physics and computing.
4.1 Definitions and Concepts
When a matrix \(M\) transforms a vector \(\mathbf{e}\), the transformed vector \(M\mathbf{e}\) usually points in a completely different direction.
However, certain special vectors, called Eigenvectors, only get scaled (stretched or compressed) by the transformation. They do not change direction (or they change to the opposite direction, which is scaling by a negative number).
- An Eigenvector \(\mathbf{e}\) is a non-zero vector such that when multiplied by the matrix \(M\), the result is a scalar multiple of \(\mathbf{e}\).
- The Eigenvalue, \(\lambda\) (lambda), is the scalar factor by which the eigenvector is scaled.
This relationship is summarized by the Eigenvalue Equation:
$$M\mathbf{e} = \lambda \mathbf{e}$$
Syllabus Note: You will only deal with real eigenvalues in this course, although complex eigenvalues exist. Repeated eigenvalues are allowed.
4.2 Finding Eigenvalues: The Characteristic Equation
To find the eigenvalues \(\lambda\), we must rearrange the eigenvalue equation:
$$M\mathbf{e} = \lambda I \mathbf{e}$$ $$M\mathbf{e} - \lambda I \mathbf{e} = \mathbf{0}$$ $$(M - \lambda I)\mathbf{e} = \mathbf{0}$$Since \(\mathbf{e}\) (the eigenvector) is defined as a non-zero vector, this system of equations only has a non-trivial solution if the matrix \((M - \lambda I)\) is singular.
Therefore, eigenvalues \(\lambda\) are the solutions to the Characteristic Equation:
$$\det(M - \lambda I) = 0$$
Step-by-Step for \(2 \times 2\) or \(3 \times 3\) Matrices:
- Form the matrix \((M - \lambda I)\) by subtracting \(\lambda\) from every element on the main diagonal of \(M\).
- Calculate the determinant of \((M - \lambda I)\).
- Set the determinant equal to zero: \(\det(M - \lambda I) = 0\).
- Solve the resulting polynomial (quadratic for \(2 \times 2\), cubic for \(3 \times 3\)) to find the values of \(\lambda\) (the eigenvalues).
4.3 Finding Eigenvectors
Once you have found an eigenvalue \(\lambda\), you find the corresponding eigenvector \(\mathbf{e}\) by substituting \(\lambda\) back into the equation:
$$(M - \lambda I)\mathbf{e} = \mathbf{0}$$
This yields a system of simultaneous equations. Because \(\det(M - \lambda I) = 0\), the equations will be linearly dependent, meaning you will find a whole line of possible vectors (an eigenvector can be scaled by any constant). You typically choose a simple integer form for the eigenvector.
Common Mistake: Accidentally setting \(\mathbf{e} = \mathbf{0}\). Remember, eigenvectors must be non-zero!
Quick Takeaway for Section 4: Eigenvectors are the special directions scaled by eigenvalues \(\lambda\). Find \(\lambda\) first using the characteristic equation \(\det(M - \lambda I) = 0\).