A2 October 2020 Paper 1 Q3
3 You are given the matrix \(\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix}\).
(a) Find \(\mathbf{A}^4\). [1]
(b) Describe the transformation that \(\mathbf{A}\) represents. [2]
The matrix \(\mathbf{B}\) represents a reflection in the plane \(x = 0\).
(c) Write down the matrix \(\mathbf{B}\). [1]
The point \(P\) has coordinates \((2, 3, 4)\). The point \(P'\) is the image of \(P\) under the transformation represented by \(\mathbf{B}\).
(d) Find the coordinates of \(P'\). [1]
| Scheme | Marks | AO |
|---|---|---|
| \(\mathbf{A}^4 = \mathbf{I}\) | B1 | 1.1 |
| [1] |
Notes
B1: Accept \(3 \times 3\) matrix
| Scheme | Marks | AO |
|---|---|---|
| Rotation | B1 | 2.2a |
| Clockwise \(90^\circ\) about \(x\)-axis | B1 | 2.2a |
| [2] |
Notes
B1: Or \(270^\circ\) anticlockwise. Accept radians
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) | B1 | 1.1 |
| [1] |
Notes
B1: All correct
| Scheme | Marks | AO |
|---|---|---|
| \((-2, 3, 4)\) | B1 | 1.1 |
| [1] |
Notes
B1: \(\begin{pmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} = \begin{pmatrix} -2 \\ 3 \\ 4 \end{pmatrix}\)
Allow vector as answer