Is matrix multiplication commutative?

No: in general \mathbf{AB}\ne\mathbf{BA}.

What property lets you write \mathbf{ABC} without brackets?

Associativity: (\mathbf{AB})\mathbf{C}=\mathbf{A}(\mathbf{BC}).

State the 2\times2 identity matrix.

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

What does \mathbf{AI} equal?

\mathbf{A} (and \mathbf{IA}=\mathbf{A} too).

Matrix for anticlockwise rotation through \theta about the origin?

\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\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 a stretch of scale factor k parallel to the y-axis?

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

Matrix for an enlargement, scale factor k, centre the origin?

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

What transformation does \mathbf{AB} represent?

\mathbf{B} followed by \mathbf{A}.

Matrix for reflection in the plane z=0?

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

Matrix for rotation through \theta about the z-axis?

\begin{pmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\0&0&1\end{pmatrix}

Matrix algebra and linear transformationsEdexcel A-Level Further Maths: Flashcards

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When are $\mathbf{A}$ and $\mathbf{B}$ conformable for $\mathbf{AB}$?

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When are $\mathbf{A}$ and $\mathbf{B}$ conformable for $\mathbf{AB}$ ?: When the number of columns of $\mathbf{A}$ equals the number of rows of $\mathbf{B}$ .
Is matrix multiplication commutative?: No: in general $\mathbf{AB}\ne\mathbf{BA}$ .
What property lets you write $\mathbf{ABC}$ without brackets?: Associativity: $(\mathbf{AB})\mathbf{C}=\mathbf{A}(\mathbf{BC})$ .
State the $2\times2$ identity matrix.: $\begin{pmatrix}1&0\\0&1\end{pmatrix}$
What does $\mathbf{AI}$ equal?: $\mathbf{A}$ (and $\mathbf{IA}=\mathbf{A}$ too).
What do the columns of a transformation matrix represent?: The images of $(1,0)$ and $(0,1)$ .
Matrix for anticlockwise rotation through $\theta$ about the origin?: $\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}$
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 a stretch of scale factor $k$ parallel to the $y$ -axis?: $\begin{pmatrix}1&0\\0&k\end{pmatrix}$
Matrix for an enlargement, scale factor $k$ , centre the origin?: $\begin{pmatrix}k&0\\0&k\end{pmatrix}$
What transformation does $\mathbf{AB}$ represent?: $\mathbf{B}$ followed by $\mathbf{A}$ .
Matrix for reflection in the plane $z=0$ ?: $\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix}$
Matrix for rotation through $\theta$ about the $z$ -axis?: $\begin{pmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\0&0&1\end{pmatrix}$

Exam questions on Matrix algebra and linear transformations

$\mathbf{A}=\begin{pmatrix}2&1\\0&3\end{pmatrix}$ and $\mathbf{B}=\begin{pmatrix}1&-1\\2&0\end{pmatrix}$ .
Find the matrix $\mathbf{C}$ such that $\mathbf{A}+\mathbf{C}=2\mathbf{B}$ .2 marks
Transformation $\mathrm{P}$ is a reflection in the line $y=x$ . Transformation $\mathrm{Q}$ is a rotation through $90^\circ$ anticlockwise about the origin. The matrix $\mathbf{M}$ represents $\mathrm{P}$ followed by $\mathrm{Q}$ .
Find $\mathbf{M}$ and describe fully the single transformation represented by $\mathbf{M}$ .2 marks
The matrix $\mathbf{T}$ represents an enlargement with scale factor $3$ and centre the origin, followed by a stretch with scale factor $2$ parallel to the $x$ -axis.
Find $\mathbf{T}$ .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).