A2 June 2019 Paper 1 Q7

AQACurrent spec4 marksMatrices

7 Three non-singular square matrices, \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{R}\) are such that

\[\mathbf{AR} = \mathbf{B}\]

The matrix \(\mathbf{R}\) represents a rotation about the \(z\)-axis through an angle \(\theta\) and

\[\mathbf{B} = \begin{bmatrix} -\cos\theta & \sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}\]

(a) Show that \(\mathbf{A}\) is independent of the value of \(\theta\). [3 marks]

(b) Give a full description of the single transformation represented by the matrix \(\mathbf{A}\). [1 mark]

Mark scheme (a)

Scheme	Marks	AO
Finds the correct matrix for \(\mathbf{R}^{-1}\) PI by correct \(\mathbf{A}\)	B1	2.2a
Appropriate method to find \(\mathbf{A}\), such as post multiplying \(\mathbf{B}\) by \(\mathbf{R}^{-1}\) PI by correct \(\mathbf{A}\)	M1	1.1a
Completes a rigorous argument to show the required result, including finding the correct matrix for \(\mathbf{A}\). Must include conclusion that \(\mathbf{A}\) is independent of \(\theta\)	R1	2.1

Typical solution

\[\mathbf{R}^{-1} = \begin{bmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}\]\[\mathbf{A} = \mathbf{BR}^{-1}\]\[\mathbf{A} = \begin{bmatrix} -\cos\theta & \sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix}\]\[\mathbf{A} = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\]

\(\mathbf{A}\) is independent of \(\theta\).

Scheme	Marks	AO
States fully correct (single) geometrical description. Eg Reflection in \(y\)/\(z\) plane.	E1	3.2a
	(4 marks)