Challenging +1.2 This question requires finding eigenvalues of a 2×2 matrix (using the characteristic equation) and interpreting their nature geometrically. Students must recognize that invariant lines exist only when real eigenvalues exist. The calculation is straightforward (determinant of λI - A = 0 gives λ² - 7λ + 22 = 0, discriminant = -39 < 0), but connecting complex eigenvalues to 'no invariant lines' requires conceptual understanding beyond routine computation, making it moderately above average difficulty for Further Maths.
$$\mathbf{A} = \begin{pmatrix} 4 & -2 \\ 5 & 3 \end{pmatrix}$$
The matrix \(\mathbf{A}\) represents the linear transformation \(M\).
Prove that, for the linear transformation \(M\), there are no invariant lines. [5]
$$\mathbf{A} = \begin{pmatrix} 4 & -2 \\ 5 & 3 \end{pmatrix}$$
The matrix $\mathbf{A}$ represents the linear transformation $M$.
Prove that, for the linear transformation $M$, there are no invariant lines. [5]
\hfill \mbox{\textit{SPS SPS FM 2023 Q4 [5]}}