Matrices

flowchart TD
    A[02 Matrices] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Matrices provide a compact algebraic framework for representing and manipulating systems of linear Equations, geometric transformations, and, at a more advanced level, quantum mechanical states and Data structures. This topic develops the algebra of matrices and their interpretation as linear Transformations of the plane and space.

Definition. An $m \times n$ matrix is a rectangular array of numbers arranged in $m$ rows and $n$ columns. We write $\mathbf{A} = (a_{ij})$ where $a_{ij}$ is the entry in row $i$ Column $j$. The set of all $m \times n$ matrices with real entries is denoted $M_{m\times n}(\mathbb{R})$.

A matrix with a single row is a row vector, and a matrix with a single column is a column Vector.

For matrices $\mathbf{A}, \mathbf{B} \in M_{m \times n}(\mathbb{R})$ and scalar $k \in \mathbb{R}$:

Matrix addition is commutative ($\mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}$) and Associative.

Definition. If $\mathbf{A} \in M_{m \times p}(\mathbb{R})$ and $\mathbf{B} \in M_{p \times n}(\mathbb{R})$ The product $\mathbf{AB} \in M_{m \times n}(\mathbb{R})$ is defined by:

Matrix multiplication is associative but not commutative : $\mathbf{AB} \neq \mathbf{BA}$.

The $n \times n$ identity matrix $\mathbf{I}_n$ satisfies $\mathbf{A}\mathbf{I}_n = \mathbf{I}_n\mathbf{A} = \mathbf{A}$ for any $\mathbf{A} \in M_{m \times n}(\mathbb{R})$.

Definition. For $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$:

Definition. For $\mathbf{A} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$ the determinant is computed by cofactor expansion along any row or column:

The signs alternate $+$, $-$, $+$ along the first row (following the checkerboard pattern).

For $n \times n$ matrices $\mathbf{A}, \mathbf{B}$ and scalar $k$:

1. $\det(\mathbf{I}_n) = 1$
2. $\det(\mathbf{A}^T) = \det(\mathbf{A})$
3. $\det(k\mathbf{A}) = k^n \det(\mathbf{A})$
4. Swapping two rows (or columns) multiplies the determinant by $-1$
5. A matrix with a zero row (or column) has $\det = 0$
6. $\det(\mathbf{A}) = 0$ if and only if $\mathbf{A}$ is singular (non-invertible)

Let $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} e & f \\ g & h \end{pmatrix}$.

Intuition. The determinant measures how a matrix scales area (in 2D) or volume (in 3D). Composing two transformations multiplies their area/volume scaling factors, which is why Determinants multiply.

Definition. The inverse of a square matrix $\mathbf{A}$ Written $\mathbf{A}^{-1}$ Is the Unique matrix satisfying:

An inverse exists if and only if $\det(\mathbf{A}) \neq 0$. Such a matrix is called non-singular.

For $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with $\det(\mathbf{A}) = ad - bc \neq 0$:

For a $3 \times 3$ matrix $\mathbf{A}$:

Where the adjugate (or adjoint) matrix $\operatorname{adj}(\mathbf{A})$ is the transpose of the cofactor matrix.

Definition. The cofactor $C_{ij}$ of entry $a_{ij}$ is $(-1)^{i+j}$ times the determinant of The submatrix obtained by deleting row $i$ and column $j$. The cofactor matrix has entries $C_{ij}$ And $\operatorname{adj}(\mathbf{A}) = (C_{ij})^T$.

Suppose $\mathbf{B}$ and $\mathbf{C}$ are both inverses of $\mathbf{A}$. Then:

A system of $n$ linear equations in $n$ unknowns can be written as $\mathbf{A}\mathbf{x} = \mathbf{b}$ where $\mathbf{A}$ is the coefficient matrix, $\mathbf{x}$ is the column vector of unknowns, and $\mathbf{b}$ is the column vector of constants.

If $\mathbf{A}$ is non-singular, the unique solution is:

For a $2 \times 2$ system:

• $\det(\mathbf{A}) \neq 0$: the two lines intersect at a unique point.
• $\det(\mathbf{A}) = 0$ and the equations are consistent: the lines are coincident (infinitely many solutions).
• $\det(\mathbf{A}) = 0$ and the equations are inconsistent: the lines are parallel (no solutions).

For a system $\mathbf{A}\mathbf{x} = \mathbf{b}$ where $\det(\mathbf{A}) \neq 0$:

Where $\mathbf{A}_i$ is $\mathbf{A}$ with column $i$ replaced by $\mathbf{b}$.

A $2 \times 2$ matrix $\mathbf{T}$ represents a linear transformation of $\mathbb{R}^2$: the point $(x, y)$ is mapped to $(x', y')$ where $\begin{pmatrix} x' \\ y' \end{pmatrix} = \mathbf{T}\begin{pmatrix} x \\ y \end{pmatrix}$.

Reflection in the $x$-axis: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

Reflection in the $y$-axis: $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$

Reflection in the line $y = x$: $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

Reflection in the line $y = \tan\theta\, x$:

The reflection of a vector in a line through the origin making angle $\theta$ with the $x$-axis can be decomposed: first rotate by $-\theta$ to align the mirror with the $x$-axis, reflect in the $x$-axis, then rotate back by $\theta$.

Rotation anticlockwise by angle $\theta$ about the origin:

Note: $\det(\mathbf{R}_\theta) = \cos^2\theta + \sin^2\theta = 1$.

Enlargement by scale factor $k$ about the origin: $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$

$\det = k^2$ Confirming the area is scaled by $k^2$.

Shear parallel to the $x$-axis by factor $k$: $\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$

Shear parallel to the $y$-axis by factor $k$: $\begin{pmatrix} 1 & 0 \\ k & 1 \end{pmatrix}$

Note: $\det = 1$ for shears, so area is preserved.

If transformation $\mathbf{A}$ is followed by transformation $\mathbf{B}$ The combined Transformation is represented by $\mathbf{BA}$ (rightmost applied first).

Rotation about the $x$-axis by angle $\theta$:

Rotation about the $y$-axis by angle $\theta$:

Rotation about the $z$-axis by angle $\theta$:

Each has $\det = 1$ and represents a rigid motion preserving distances and orientation.

Reflection in the plane $x = 0$ (the $yz$-plane): $\begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

Reflection in the plane $y = 0$ (the $xz$-plane): $\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

Reflection in the plane $z = 0$ (the $xy$-plane): $\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}$

Each has $\det = -1$ Confirming orientation reversal.

Info: info require this topic at A Level. Definition. Let $\mathbf{A}$ be an $n \times n$ matrix. A scalar $\lambda$ is an eigenvalue of $\mathbf{A}$ if there exists a non-zero vector $\mathbf{v}$ such that:

The vector $\mathbf{v}$ is called an eigenvector corresponding to $\lambda$.

$\mathbf{A}\mathbf{v} = \lambda\mathbf{v} \iff (\mathbf{A} - \lambda\mathbf{I})\mathbf{v} = \mathbf{0}$.

For a non-trivial solution ($\mathbf{v} \neq \mathbf{0}$), we require $\det(\mathbf{A} - \lambda\mathbf{I}) = 0$.

This is the characteristic equation of $\mathbf{A}$. Its roots are the eigenvalues.

For each eigenvalue $\lambda$ Solve $(\mathbf{A} - \lambda\mathbf{I})\mathbf{v} = \mathbf{0}$ by Row reduction.

Definition. A matrix $\mathbf{A}$ is diagonalisable if there exists an invertible matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that:

The columns of $\mathbf{P}$ are the eigenvectors of $\mathbf{A}$ And the diagonal entries of $\mathbf{D}$ are the corresponding eigenvalues.

A matrix is diagonalisable if and only if it has $n$ linearly independent eigenvectors (always true for $n$ distinct eigenvalues).

Intuition. Diagonalisation changes to a coordinate system where the transformation acts Independently on each axis (stretching by eigenvalues). In this basis, the matrix takes its simplest Possible form.

Problem 1. Given $\mathbf{A} = \begin{pmatrix} 3 & -1 \\ 2 & 4 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} 1 & 5 \\ -2 & 0 \end{pmatrix}$ Compute $\mathbf{AB} - \mathbf{BA}$.

Problem 2. Find the determinant and inverse of $\mathbf{A} = \begin{pmatrix} 2 & 1 & 3 \\ 0 & -1 & 2 \\ 1 & 0 & 1 \end{pmatrix}$.

Problem 3. Find the matrix representing a reflection in the line $y = \sqrt{3}\,x$ And verify That $\mathbf{M}^2 = \mathbf{I}$.

Problem 4. The triangle with vertices (0, 0)$$(2, 0)$$(0, 1) is transformed by the matrix $\mathbf{T} = \begin{pmatrix} 3 & 1 \\ 0 & 2 \end{pmatrix}$. Find the coordinates of the vertices of the image, and verify that the area scales by $|\det(\mathbf{T})|$.

Problem 5. Solve the system of equations using matrices:

Problem 6. Find the single $2 \times 2$ matrix that represents an enlargement by scale factor 2 About the origin followed by a rotation of $90^\circ$ anticlockwise. Show that this is equivalent to A single rotation of $90^\circ$ combined with an enlargement by factor 2.

Problem 7. AQA only. Find the eigenvalues and corresponding eigenvectors of $\mathbf{A} = \begin{pmatrix} 5 & 4 \\ 1 & 2 \end{pmatrix}$ And write down a matrix $\mathbf{P}$ And diagonal matrix $\mathbf{D}$ such that $\mathbf{A} = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}$.

Problem 8. AQA only. The matrix $\mathbf{A} = \begin{pmatrix} 3 & -2 \\ 1 & 0 \end{pmatrix}$ has Eigenvalues 1 and 2. Use this to compute $\mathbf{A}^5$ without multiplying matrices five times.