| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find line of invariant points |
| Difficulty | Standard +0.8 This is a Further Maths question requiring understanding of self-inverse matrices (A² = I), solving simultaneous equations from matrix multiplication, and finding invariant points (eigenvector analysis). While the individual techniques are standard, combining them requires solid conceptual understanding and multi-step algebraic manipulation, placing it moderately above average difficulty. |
| Spec | 4.03g Invariant points and lines4.03o Inverse 3x3 matrix |
$$\mathbf{A} = \begin{pmatrix} 2 & a \\ a-4 & b \end{pmatrix}$$
where $a$ and $b$ are non-zero constants.
Given that the matrix $\mathbf{A}$ is self-inverse,
\begin{enumerate}[label=(\alph*)]
\item determine the value of $b$ and the possible values for $a$.
[5]
The matrix $\mathbf{A}$ represents a linear transformation $M$.
Using the smaller value of $a$ from part (a),
\item show that the invariant points of the linear transformation $M$ form a line, stating the equation of this line.
[3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q13 [8]}}