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

1. 1
   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 $x=0$.

   (a)

   Find the image of the point $(1,2,3)$ under the transformation $\mathbf{P}$.

   [1 mark]

   • A$(2,-1,3)$
   • B$(-2,1,3)$
   • C$(-1,-2,3)$
   • D$(2,1,3)$

   (b)

   Which matrix represents $\mathbf{P}$ followed by $\mathbf{Q}$?

   [1 mark]

   • A$\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$
   • B$\begin{pmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$
   • C$\begin{pmatrix} -1 & -1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}$
   • D$\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}$

   (c)

   Write down the matrix that represents the inverse of the transformation $\mathbf{P}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The matrix $\mathbf{A}=\begin{pmatrix} 1 & 2 & 0 \\ 3 & k & -1 \\ 2 & 0 & 1 \end{pmatrix}$, where $k$ is a constant.

   (a)

   Find $\det\mathbf{A}$ in terms of $k$.

   [1 mark]

   • A$10-k$
   • B$k+10$
   • C$k-5$
   • D$k-10$

   (b)

   For what value of $k$ is $\mathbf{A}$ singular?

   [1 mark]

   • A$k=0$
   • B$k=-10$
   • C$k=10$
   • D$k=5$

   (c)

   Given that $k=4$, write down $\mathbf{A}^{\mathrm{T}}$ and find $\det\mathbf{A}^{\mathrm{T}}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   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 $x=y$.

   (a)

   Find $\mathbf{M}^{-1}$.

   [3 marks]

   (b)

   Find $(\mathbf{MN})^{-1}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The matrix $\mathbf{A}=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}$ represents an anticlockwise rotation of $90^\circ$ about the $x$-axis and $\mathbf{B}=\begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1 \end{pmatrix}$ represents a stretch of scale factor $2$ parallel to the $x$-axis and to the $y$-axis combined with a reflection in the plane $z=0$. The transformation $T$ is $\mathbf{A}$ followed by $\mathbf{B}$.

   (a)

   Find the matrix $\mathbf{T}$ that represents $T$, find $\det\mathbf{T}$, and find $\mathbf{T}^{-1}$.

   [6 marks]

   (b)

   Describe the transformation represented by $\mathbf{T}^{-1}$ as two transformations, stating their order. Find the coordinates of the point whose image under $T$ is $(6,-4,-3)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions