Matrices and linear transformations (WACE Year 12 Mathematics Specialist Unit 4)

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.

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.

Identifying a transformation from its matrix

The reverse skill is also examined: given a matrix, name the transformation. Compare the entries with the standard templates. A matrix of the form $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ with determinant $+1$ is a rotation by $\theta$. A symmetric matrix of the form $\begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}$ with determinant $-1$ is a reflection. A diagonal matrix is a dilation, and a matrix with a single off-diagonal entry is a shear. Checking the determinant first tells you whether orientation is preserved (positive) or reversed (negative), which immediately narrows the possibilities.