Standard +0.8 This is a multi-part Further Maths question requiring eigenvalue/eigenvector computation, matrix diagonalization, and limit analysis involving powers of matrices. While eigenvalues are given (reducing computational burden), students must find eigenvectors, construct the diagonalization, and apply conceptual understanding of matrix powers and limits. The final part requires insight that |k·λ| < 1 for all eigenvalues. This is moderately challenging for Further Maths, above average difficulty overall.
9 The matrix
$$\mathbf { A } = \left( \begin{array} { r r r }
3 & 1 & 4 \\
1 & 5 & - 1 \\
2 & 1 & 5
\end{array} \right)$$
has eigenvalues \(1,5,7\). Find a set of corresponding eigenvectors.
Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } ^ { - 1 }\). [0pt]
[The evaluation of \(\mathbf { P } ^ { - 1 }\) is not required.]
Determine the set of values of the real constant \(k\) such that \(k ^ { n } \mathbf { A } ^ { n }\) tends to the zero matrix as \(n \rightarrow \infty\).
Exhibits \(A - \lambda I\) in a numerical form, where \(\lambda = 1, 5\) or \(7\)
M1
Uses this form in some way to obtain I eigenvector
M1
Eigenvectors: any non-zero scaling of \(\begin{pmatrix}17\\-6\\-7\end{pmatrix}\), \(\begin{pmatrix}1\\-2\\1\end{pmatrix}\), \(\begin{pmatrix}1\\0\\1\end{pmatrix}\)
A1A1A1
\(\mathbf{P} = \begin{pmatrix}17 & 1 & 1\\-6 & -2 & 0\\-7 & 1 & 1\end{pmatrix}\) Do not allow \(\begin{pmatrix}0\\0\\0\end{pmatrix}\)
9 The matrix
$$\mathbf { A } = \left( \begin{array} { r r r }
3 & 1 & 4 \\
1 & 5 & - 1 \\
2 & 1 & 5
\end{array} \right)$$
has eigenvalues $1,5,7$. Find a set of corresponding eigenvectors.
Find a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { A } ^ { n } = \mathbf { P D P } ^ { - 1 }$.\\[0pt]
[The evaluation of $\mathbf { P } ^ { - 1 }$ is not required.]\\
Determine the set of values of the real constant $k$ such that $k ^ { n } \mathbf { A } ^ { n }$ tends to the zero matrix as $n \rightarrow \infty$.
\hfill \mbox{\textit{CAIE FP1 2009 Q9 [11]}}