Standard +0.8 This is a standard Further Maths eigenvalue/diagonalization question with eigenvalues given, requiring routine eigenvector calculation, matrix diagonalization, and a limit involving powers of matrices. The final limit calculation requires recognizing that only the largest eigenvalue (4) survives when factored out, which is a moderately sophisticated insight beyond pure mechanical computation. The multi-step nature and the conceptual understanding needed for the limit pushes this slightly above average difficulty for Further Maths content.
It is given that the eigenvalues of the matrix \(\mathbf{M}\), where
$$\mathbf{M} = \begin{pmatrix} 4 & 1 & -1 \\ -4 & -1 & 4 \\ 0 & -1 & 5 \end{pmatrix},$$
are \(1, 3, 4\). Find a set of corresponding eigenvectors. [4]
Write down a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that
$$\mathbf{M}^n = \mathbf{P}\mathbf{D}^n\mathbf{P}^{-1},$$
where \(n\) is a positive integer. [2]
Find \(\mathbf{P}^{-1}\) and deduce that
$$\lim_{n \to \infty} 4^{-n}\mathbf{M}^n = \begin{pmatrix} -\frac{1}{3} & 0 & -\frac{1}{3} \\ \frac{4}{3} & 0 & \frac{4}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} \end{pmatrix}.$$ [5]
It is given that the eigenvalues of the matrix $\mathbf{M}$, where
$$\mathbf{M} = \begin{pmatrix} 4 & 1 & -1 \\ -4 & -1 & 4 \\ 0 & -1 & 5 \end{pmatrix},$$
are $1, 3, 4$. Find a set of corresponding eigenvectors. [4]
Write down a matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that
$$\mathbf{M}^n = \mathbf{P}\mathbf{D}^n\mathbf{P}^{-1},$$
where $n$ is a positive integer. [2]
Find $\mathbf{P}^{-1}$ and deduce that
$$\lim_{n \to \infty} 4^{-n}\mathbf{M}^n = \begin{pmatrix} -\frac{1}{3} & 0 & -\frac{1}{3} \\ \frac{4}{3} & 0 & \frac{4}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} \end{pmatrix}.$$ [5]
\hfill \mbox{\textit{CAIE FP1 2005 Q10 [11]}}