Year: 2021
Paper: 2
Question Number: 7
Course: LFM Pure
Section: Linear transformations
No solution available for this problem.
Candidates were generally well prepared for many of the questions on this paper, with the questions requiring more standard operations seeing the greatest levels of success. Candidates need to ensure that solutions to the questions are supported by sufficient evidence of the mathematical steps, for example when proving a given result or deducing the properties of graphs that are to be sketched. In a significant number of steps there were marks lost through simple errors such as mistakes in arithmetic or confusion of sine and cosine functions, so it is important for candidates to maintain accuracy in their solutions to these questions.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item The matrix $\mathbf{R}$ represents an anticlockwise rotation through angle $\varphi$ ($0^\circ \leqslant \varphi < 360^\circ$) in two dimensions, and the matrix $\mathbf{R} + \mathbf{I}$ also represents a rotation in two dimensions. Determine the possible values of $\varphi$ and deduce that $\mathbf{R}^3 = \mathbf{I}$.
\item Let $\mathbf{S}$ be a real matrix with $\mathbf{S}^3 = \mathbf{I}$, but $\mathbf{S} \neq \mathbf{I}$.
Show that $\det(\mathbf{S}) = 1$.
Given that
\[
\mathbf{S} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]
show that $\mathbf{S}^2 = (a+d)\mathbf{S} - \mathbf{I}$.
Hence prove that $a + d = -1$.
\item Let $\mathbf{S}$ be a real $2 \times 2$ matrix.
Show that if $\mathbf{S}^3 = \mathbf{I}$ and $\mathbf{S} + \mathbf{I}$ represents a rotation, then $\mathbf{S}$ also represents a rotation.
What are the possible angles of the rotation represented by $\mathbf{S}$?
\end{questionparts}
Solutions to this question often highlighted a number of issues with understanding of matrices. For example, some candidates thought that, if the product of two matrices is zero, then one of the two matrices must be zero. Similarly, some solutions treated the number 1 and the identity matrix as interchangeable. There were also many poor examples of manipulation of determinants seen. Candidates were able to engage well with part (i), although perfect solutions to this part were uncommon. In part (ii) many candidates were able to show the given result successfully. However, a number of attempts at this part of the question made the assumption that the matrix was a rotation, even though this is not given in the question. In part (iii) there were a number of solutions that assumed that the determinant being 1 was a sufficient condition for the matrix to represent a rotation or gave an insufficient justification that the matrix represents a rotation. Many candidates were able to deduce the angles of the rotation correctly in this part.