Use Cayley-Hamilton for matrix power

2 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 2 & 12 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{array} \right) .$$ Use the characteristic equation of \(\mathbf { A }\) to show that $$\mathbf { A } ^ { 4 } = p \mathbf { A } ^ { 2 } + q \mathbf { l }$$ where \(p\) and \(q\) are integers to be determined.

3

1. 1. A \(3 \times 3\) matrix \(\mathbf { M }\) has characteristic equation $$2 \lambda ^ { 3 } + \lambda ^ { 2 } - 13 \lambda + 6 = 0$$ Show that \(\lambda = 2\) is an eigenvalue of \(\mathbf { M }\). Find the other eigenvalues.
   2. An eigenvector corresponding to \(\lambda = 2\) is \(\left( \begin{array} { r } 3 \\ - 3 \\ 1 \end{array} \right)\). Evaluate \(\mathbf { M } \left( \begin{array} { r } 3 \\ - 3 \\ 1 \end{array} \right)\) and \(\mathbf { M } ^ { 2 } \left( \begin{array} { r } 1 \\ - 1 \\ \frac { 1 } { 3 } \end{array} \right)\).
      Solve the equation \(\mathbf { M } \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } 3 \\ - 3 \\ 1 \end{array} \right)\).
   3. Find constants \(A , B , C\) such that $$\mathbf { M } ^ { 4 } = A \mathbf { M } ^ { 2 } + B \mathbf { M } + C \mathbf { I }$$
2. A \(2 \times 2\) matrix \(\mathbf { N }\) has eigenvalues -1 and 2, with eigenvectors \(\binom { 1 } { 2 }\) and \(\binom { - 1 } { 1 }\) respectively. Find \(\mathbf { N }\). Section B (18 marks)

5 The matrix \(\mathbf { P }\) is given by \(\mathbf { P } = \left( \begin{array} { l l } a & 0 \\ 2 & 3 \end{array} \right)\) where \(a\) is a constant and \(a \neq 3\).

1. Given that the acute angle between the directions of the eigenvectors of \(\mathbf { P }\) is \(\frac { 1 } { 4 } \pi\) radians, determine the possible values of \(a\).
2. You are given instead that \(\mathbf { P }\) satisfies the matrix equation \(\mathbf { I } = \mathbf { P } ^ { 2 } + r \mathbf { P }\) for some rational number \(r\).
   1. Use the Cayley-Hamilton theorem to determine the value of \(a\) and the corresponding value of \(r\).
   2. Hence show that \(\mathbf { P } ^ { 4 } = \mathbf { s } \mathbf { + t } \mathbf { t } \mathbf { P }\) where \(s\) and \(t\) are rational numbers to be determined. You should not calculate \(\mathbf { P } ^ { 4 }\).

$$A = \left( \begin{array} { r r } 1 & - 2 \\ 1 & 4 \end{array} \right)$$

1. Show that the characteristic equation for \(\mathbf { A }\) is \(\lambda ^ { 2 } - 5 \lambda + 6 = 0\)
2. Use the Cayley-Hamilton theorem to find integers \(p\) and \(q\) such that $$\mathbf { A } ^ { 3 } = p \mathbf { A } + q \mathbf { I }$$ (ii) Given that the \(2 \times 2\) matrix \(\mathbf { M }\) has eigenvalues \(- 1 + \mathrm { i }\) and \(- 1 - \mathrm { i }\), with eigenvectors \(\binom { 1 } { 2 - \mathrm { i } }\) and \(\binom { 1 } { 2 + \mathrm { i } }\) respectively, find the matrix \(\mathbf { M }\).

1. Given that

$$A = \left( \begin{array} { l l } 3 & 1 \\ 6 & 4 \end{array} \right)$$

1. find the characteristic equation of the matrix \(\mathbf { A }\).
2. Hence show that \(\mathbf { A } ^ { 3 } = 43 \mathbf { A } - 42 \mathbf { I }\).

1. $$\mathbf { A } = \left( \begin{array} { r r } - 1 & a \\ 3 & 8 \end{array} \right)$$ where \(a\) is a constant.

1. Determine, in expanded form in terms of \(a\), the characteristic equation for \(\mathbf { A }\).
2. Hence use the Cayley-Hamilton theorem to determine values of \(a\) and \(b\) such that $$\mathbf { A } ^ { 3 } = \mathbf { A } + b \mathbf { I }$$ where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.