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

Exam questions on Three-dimensional transformations and 3x3 matrices

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 $x=0$.
   Write down the matrix that represents the inverse of the transformation $\mathbf{P}$.2 marks
2. 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
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 $x=y$.
   Find $\mathbf{M}^{-1}$.3 marks