Matrix arithmetic, determinants, inverses of 2x2 and 3x3 matrices, matrices as linear transformations, invariant points and lines, and solving linear systems.

What this dot point is asking

Edexcel wants you to add, subtract and multiply matrices, evaluate $2 \times 2$ and $3 \times 3$ determinants, find inverses, interpret matrices as linear transformations of the plane and of space, find invariant points and invariant lines, and use the inverse matrix to solve simultaneous equations. Matrices are a guaranteed Core Pure topic and frequently combine with transformation geometry and, in the $3 \times 3$ case, with solving systems of three equations.

Arithmetic and determinants

Matrix addition is component-wise and only defined for matrices of the same size. Multiplication combines each row of the first matrix with each column of the second, so $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$. The $(i, j)$ entry of $AB$ is the dot product of row $i$ of $A$ with column $j$ of $B$. Order matters: $AB$ and $BA$ are usually different and may not even both exist.

The determinant is the single most important number attached to a square matrix because it decides invertibility and gives the area (or volume) scale factor of the transformation.

A matrix with $\det A = 0$ is called singular and has no inverse; geometrically it collapses the plane onto a line (or space onto a plane), losing a dimension.

Inverting a 3x3 matrix

To invert a non-singular $3 \times 3$ matrix, follow four steps: find the determinant; build the matrix of minors (each entry is the $2 \times 2$ determinant left after deleting that entry's row and column); apply the chequerboard sign pattern $\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}$ to get the cofactor matrix; transpose to get the adjugate; then divide by the determinant.

Matrices as transformations

A $2 \times 2$ matrix maps the plane linearly, fixing the origin. The columns are the images of the basis vectors $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$, so you can read a transformation straight off the matrix. Standard cases are the rotation through $\theta$ anticlockwise about the origin, $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$; the reflection in the line $y = x\tan\theta$; and the enlargement by scale factor $k$, $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$. A composition of transformations is the product of their matrices, applied right to left, so "first $P$ then $Q$" is the matrix $QP$.

To find invariant points, solve $(A - I)\mathbf{x} = \mathbf{0}$. To find invariant lines $y = mx + c$, demand that the image of a general point on the line also satisfies the same line equation, then equate gradients and intercepts.

Solving linear systems

A system written as $A\mathbf{x} = \mathbf{b}$ has the unique solution $\mathbf{x} = A^{-1}\mathbf{b}$ when $\det A \neq 0$. When $\det A = 0$ the system is either inconsistent (no solution, parallel planes) or has infinitely many solutions (a line or plane of solutions), so you must check by substitution rather than assume one or the other. For three equations in three unknowns, the geometry of the three planes (meeting at a point, in a line, or not at all) mirrors these algebraic cases.

Examples in context

Matrices reach across the whole Further Maths course. The determinant as an area scale factor links to integration and to the Jacobian idea in change of variables. Eigenvalues and eigenvectors (the natural sequel) decide invariant lines through the origin and diagonalise a matrix, and the eigenvalues of a rotation matrix are the complex conjugates $\cos\theta \pm i\sin\theta$, tying matrices to complex numbers. In mechanics, transformation matrices model rigid rotations of frames. The inverse-matrix method for systems generalises to the $3 \times 3$ case that appears in coordinate-geometry questions about intersecting planes, and proof by induction is the standard tool for establishing a formula for $A^n$ once you have spotted the pattern in the first few powers.

Try this

Q1. Find the determinant of $\begin{pmatrix} 5 & 2 \\ 3 & 4 \end{pmatrix}$ and state whether it is invertible. [2 marks]

• Cue. $\det = 20 - 6 = 14 \neq 0$, so it is invertible.

Q2. Solve $\begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 5 \end{pmatrix}$ using the inverse. [4 marks]

• Cue. $\det = 5$, inverse $\frac{1}{5}\begin{pmatrix} 3 & -1 \\ -1 & 2 \end{pmatrix}$, giving $x = 2$, $y = 1$.

Q3. The matrix $\begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}$ represents a shear. Find its invariant points. [3 marks]

• Cue. Solve $(A - I)\mathbf{x} = \mathbf{0}$: $\begin{pmatrix} 0 & 3 \\ 0 & 0 \end{pmatrix}\mathbf{x} = \mathbf{0}$ gives $y = 0$, so the $x$-axis is the line of invariant points.