Matrix for reflection in the x-axis?

\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

Matrix for reflection in the y-axis?

\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}

Matrix for reflection in y=x?

\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

Matrix for reflection in y=-x?

\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}

Matrix for rotation through \theta anticlockwise about O?

\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

Matrix for rotation through 90^\circ clockwise about O?

\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}

Matrix for rotation through 180^\circ about O?

\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}

Matrix for a stretch parallel to the x-axis, scale factor k?

\begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix}

Matrix for a stretch parallel to the y-axis, scale factor k?

\begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix}

Matrix for an enlargement, centre O, scale factor k?

\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}

Which line is invariant under a stretch parallel to the x-axis?

The y-axis: points with x=0 do not move.

How do you find the image of the line y=mx+c?

Take a general point (t,mt+c), apply the matrix, then eliminate t.

Matrix representations of standard transformationsEdexcel International A Level Further Maths: Flashcards

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What do the columns of a $2\times2$ transformation matrix represent?

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What do the columns of a $2\times2$ transformation matrix represent?: The first column is the image of $(1,0)$ and the second column is the image of $(0,1)$ .
Matrix for reflection in the $x$ -axis?: $\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
Matrix for reflection in the $y$ -axis?: $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$
Matrix for reflection in $y=x$ ?: $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
Matrix for reflection in $y=-x$ ?: $\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$
Matrix for rotation through $\theta$ anticlockwise about $O$ ?: $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$
Matrix for rotation through $90^\circ$ clockwise about $O$ ?: $\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$
Matrix for rotation through $180^\circ$ about $O$ ?: $\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$
Matrix for a stretch parallel to the $x$ -axis, scale factor $k$ ?: $\begin{pmatrix} k & 0 \\ 0 & 1 \end{pmatrix}$
Matrix for a stretch parallel to the $y$ -axis, scale factor $k$ ?: $\begin{pmatrix} 1 & 0 \\ 0 & k \end{pmatrix}$
Matrix for an enlargement, centre $O$ , scale factor $k$ ?: $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$
Which line is invariant under a stretch parallel to the $x$ -axis?: The $y$ -axis: points with $x=0$ do not move.
How do you find the image of the line $y=mx+c$ ?: Take a general point $(t,mt+c)$ , apply the matrix, then eliminate $t$ .

Exam questions on Matrix representations of standard transformations

Single transformations of the plane are represented by $2\times2$ matrices acting on position vectors, with the origin $O$ as the fixed point. The point $P$ has coordinates $(3,-2)$ .
The stretch parallel to the $x$ -axis with scale factor $3$ maps $P$ to $P'$ . Write down the matrix of the stretch and find the coordinates of $P'$ .2 marks
The matrix $\mathbf{M}=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ represents a single transformation $T$ of the plane.
Show that $\mathbf{M}^2$ represents an enlargement, and state its centre and scale factor.2 marks
Triangle $T$ has vertices $A(1,1)$ , $B(3,1)$ and $C(3,4)$ .
A rotation through $60^\circ$ anticlockwise about $O$ has matrix $\mathbf{R}$ . Write down $\mathbf{R}$ with exact entries and find the exact coordinates of the image of $B$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).