| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | February |
| Marks | 10 |
| Topic | Matrices |
| Type | Invariant lines of transformation |
| Difficulty | Standard +0.3 This is a straightforward further maths matrices question with standard techniques: (a) equating matrix products requires basic multiplication, (b) is trivial—just provide any non-commutative example, (c) uses the determinant formula for area scaling, and (d) finds invariant lines by solving (B-λI)v=0 for eigenvectors. All parts are routine applications of well-known methods with no novel problem-solving required. |
| Spec | 4.03b Matrix operations: addition, multiplication, scalar4.03c Matrix multiplication: properties (associative, not commutative)4.03g Invariant points and lines4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
6.
The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $= \left( \begin{array} { l l } 1 & a \\ 3 & 0 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { l l } 4 & 2 \\ 3 & 3 \end{array} \right)$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of a such that $\mathbf { A B } = \mathbf { B A }$.
\item Prove by counter example that matrix multiplication for $2 \times 2$ matrices is not commutative.
\item A triangle of area 4 square units is transformed by the matrix $\mathbf { B }$. Find the area of the image of the triangle following this transformation.
\item Find the equations of the invariant lines of the form $y = m x$ for the transformation represented by matrix $\mathbf { B }$.\\[0pt]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2024 Q6 [10]}}