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

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

What does \mathbf{AB} mean as a transformation?

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

When is a matrix singular?

When its determinant is 0; it has no inverse.

Sign pattern for expanding a 3\times3 determinant?

\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}

How do you find \mathbf{A}^{-1} for a 3\times3 matrix?

\frac{1}{\det\mathbf{A}}\times adjugate (transpose of the cofactor matrix).

Inverse of an anticlockwise 90^\circ rotation about an axis?

A clockwise 90^\circ rotation about the same axis.

What is \det(\mathbf{AB})?

\det\mathbf{A}\times\det\mathbf{B}

Three-dimensional transformations and 3x3 matricesEdexcel International A Level Further Maths: Flashcards

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What do the columns of a transformation matrix show?

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What do the columns of a transformation matrix show?: The images of $\mathbf{i}$ , $\mathbf{j}$ and $\mathbf{k}$ .
Matrix for reflection in the plane $z=0$ ?: $\operatorname{diag}(1,1,-1)$
Matrix for anticlockwise rotation $\theta$ about the $z$ -axis?: $\begin{pmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$
What does $\mathbf{AB}$ mean as a transformation?: $\mathbf{B}$ followed by $\mathbf{A}$ .
What is $(\mathbf{AB})^{\mathrm{T}}$ ?: $\mathbf{B}^{\mathrm{T}}\mathbf{A}^{\mathrm{T}}$
What is $\det\mathbf{A}^{\mathrm{T}}$ ?: $\det\mathbf{A}$
When is a matrix singular?: When its determinant is $0$ ; it has no inverse.
Sign pattern for expanding a $3\times3$ determinant?: $\begin{pmatrix} + & - & + \\ - & + & - \\ + & - & + \end{pmatrix}$
How do you find $\mathbf{A}^{-1}$ for a $3\times3$ matrix?: $\frac{1}{\det\mathbf{A}}\times$ adjugate (transpose of the cofactor matrix).
What is $(\mathbf{AB})^{-1}$ ?: $\mathbf{B}^{-1}\mathbf{A}^{-1}$
What is the inverse of a reflection?: The same reflection.
Inverse of an anticlockwise $90^\circ$ rotation about an axis?: A clockwise $90^\circ$ rotation about the same axis.
What is $\det(\mathbf{AB})$ ?: $\det\mathbf{A}\times\det\mathbf{B}$

Exam questions on Three-dimensional transformations and 3x3 matrices

The matrix $\mathbf{P}=\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ represents an anticlockwise rotation of $90^\circ$ about the $z$ -axis, and the matrix $\mathbf{Q}=\begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ represents a reflection in the plane $x=0$ .
Write down the matrix that represents the inverse of the transformation $\mathbf{P}$ .2 marks
The matrix $\mathbf{A}=\begin{pmatrix} 1 & 2 & 0 \\ 3 & k & -1 \\ 2 & 0 & 1 \end{pmatrix}$ , where $k$ is a constant.
Given that $k=4$ , write down $\mathbf{A}^{\mathrm{T}}$ and find $\det\mathbf{A}^{\mathrm{T}}$ .2 marks
The matrix $\mathbf{M}=\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix}$ and the matrix $\mathbf{N}=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ , which represents a reflection in the plane $x=y$ .
Find $\mathbf{M}^{-1}$ .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).