Use Cayley-Hamilton for inverse

3 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { l l l } 6 & - 9 & 5 \\ 5 & - 8 & 5 \\ 1 & - 1 & 2 \end{array} \right)$$

1. Find the eigenvalues of \(\mathbf { A }\).
2. Use the characteristic equation of \(\mathbf { A }\) to show that \(\mathbf { A } ^ { - 1 } = p \mathbf { A } ^ { 2 } + q \mathbf { l }\), where \(p\) and \(q\) are constants to be determined.

3

1. Show that the characteristic equation of the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 3 & - 1 & 2 \\ - 4 & 3 & 2 \\ 2 & 1 & - 1 \end{array} \right)$$ is \(\lambda ^ { 3 } - 5 \lambda ^ { 2 } - 7 \lambda + 35 = 0\).
2. Show that \(\lambda = 5\) is an eigenvalue of \(\mathbf { M }\), and find its other eigenvalues.
3. Find an eigenvector, \(\mathbf { v }\), of unit length corresponding to \(\lambda = 5\). State the magnitudes and directions of the vectors \(\mathbf { M } ^ { 2 } \mathbf { v }\) and \(\mathbf { M } ^ { - 1 } \mathbf { v }\).
4. Use the Cayley-Hamilton theorem to find the constants \(a , b , c\) such that $$\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I } .$$ Section B (18 marks)

2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r r } 10 & 12 & - 8 \\ - 1 & 2 & 4 \\ 3 & 6 & 2 \end{array} \right)\).

1. In this question you must show detailed reasoning. Show that the characteristic equation of \(\mathbf { A }\) is \(- \lambda ^ { 3 } + 14 \lambda ^ { 2 } - 56 \lambda + 64 = 0\).
2. Use the Cayley-Hamilton theorem to determine \(\mathbf { A } ^ { - 1 }\). A matrix \(\mathbf { E }\) and a diagonal matrix \(\mathbf { D }\) are such that \(\mathbf { A } = \mathbf { E D E } ^ { - 1 }\). The elements in the diagonal of \(\mathbf { D }\) increase from top left to bottom right.
3. Determine the matrix \(\mathbf { D }\).

1. Given that

$$\mathbf { A } = \left( \begin{array} { l l } 3 & 2 \\ 2 & 2 \end{array} \right)$$

1. find the characteristic equation for the matrix \(\mathbf { A }\), simplifying your answer.
2. Hence find an expression for the matrix \(\mathbf { A } ^ { - 1 }\) in the form \(\lambda \mathbf { A } + \mu \mathbf { I }\), where \(\lambda\) and \(\mu\) are constants to be found.

3. $$\mathbf { M } = \left( \begin{array} { r r r } 1 & k & - 2 \\ 2 & - 4 & 1 \\ 1 & 2 & 3 \end{array} \right)$$ where \(k\) is a constant.

1. Show that, in terms of \(k\), a characteristic equation for \(\mathbf { M }\) is given by $$\lambda ^ { 3 } - ( 2 k + 13 ) \lambda + 5 ( k + 6 ) = 0$$ Given that \(\operatorname { det } \mathbf { M } = 5\)
2. 1. find the value of \(k\)
   2. use the Cayley-Hamilton theorem to find the inverse of \(\mathbf { M }\).

1. Matrix \(\mathbf { M }\) is given by

$$\mathbf { M } = \left( \begin{array} { r r r } 1 & 0 & a \\ - 3 & b & 1 \\ 0 & 1 & a \end{array} \right)$$ where \(a\) and \(b\) are integers, such that \(a < b\) Given that the characteristic equation for \(\mathbf { M }\) is $$\lambda ^ { 3 } - 7 \lambda ^ { 2 } + 13 \lambda + c = 0$$ where \(c\) is a constant,

1. determine the values of \(a , b\) and \(c\).
2. Hence, using the Cayley-Hamilton theorem, determine the matrix \(\mathbf { M } ^ { - 1 }\)