What is sign slips in the inverse?

The inverse swaps a and d and negates b and c. A common error is negating the wrong pair or forgetting to divide by A.

WASpecialist MathematicsSyllabus dot point

How do matrices encode and combine linear transformations of the plane?

Use 2x2 matrices for arithmetic, determinants and inverses, and as linear transformations of the plane

WACE Specialist Unit 4 matrices: 2x2 matrix arithmetic, determinants and inverses, the matrices for rotation, reflection, dilation and shear, composition of transformations by matrix multiplication, and the geometric meaning of the determinant.

Generated by Claude Opus 4.78 min answerUpdated 2026-05-29

Reviewed by: AI editorial process; not yet individually human-reviewed

Have a quick question? Jump to the Q&A page

Jump to a section

What this dot point is asking
Matrix arithmetic, determinant and inverse
Standard transformation matrices
Composition and the determinant
Solving systems with the inverse

What this dot point is asking

SCSA Unit 4 treats $2\times 2$ matrices both as objects of arithmetic and as linear transformations of the plane. You must compute products, determinants and inverses, recognise and build the standard transformation matrices, compose transformations by multiplication, and interpret the determinant geometrically.

Matrix arithmetic, determinant and inverse

For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ the determinant is $\det A = ad - bc$ . The matrix is invertible exactly when $\det A \neq 0$ , and

Matrix multiplication is associative but not commutative: in general $AB \neq BA$ . The identity is $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ , and $A A^{-1} = A^{-1} A = I$ .

Standard transformation matrices

A point $\begin{pmatrix} x \\ y \end{pmatrix}$ is mapped to $A\begin{pmatrix} x \\ y \end{pmatrix}$ . The standard matrices are:

Rotation by angle $\theta$ anticlockwise about the origin: $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ .
Reflection in the line $y = x\tan\theta$ (a line at angle $\theta$ to the $x$ -axis): $\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}$ .
Dilation with factors $k$ in $x$ and $\ell$ in $y$ : $\begin{pmatrix} k & 0 \\ 0 & \ell \end{pmatrix}$ .
Shear parallel to the $x$ -axis with factor $k$ : $\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$ .

Composition and the determinant

Applying transformation $S$ then $T$ corresponds to the product $TS$ (the first-applied matrix sits on the right, next to the vector). The determinant has a direct geometric reading.

Composing a rotation and a reflection

Reflect in the $x$ -axis, then rotate $90^\circ$ anticlockwise

Reflection in the $x$ -axis is $R = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$ . Rotation by $90^\circ$ is $T = \begin{pmatrix} \cos 90^\circ & -\sin 90^\circ \\ \sin 90^\circ & \cos 90^\circ \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ .

Apply reflection first, then rotation, so the combined matrix is $TR$ (rotation on the left).

TR = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.

The result $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ swaps $x$ and $y$ , which is reflection in the line $y = x$ . Its determinant is $(0)(0) - (1)(1) = -1$ , confirming an orientation-reversing isometry that preserves area.

Solving systems with the inverse

A linear system $A\mathbf{x} = \mathbf{b}$ with $\det A \neq 0$ has the unique solution $\mathbf{x} = A^{-1}\mathbf{b}$ . If $\det A = 0$ the system has either no solution or infinitely many.