Find P and D for A² = PDP⁻¹ or A⁻¹ = PDP⁻¹

6 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } 5 & - \frac { 22 } { 3 } & 8 \\ 0 & - 6 & 0 \\ 0 & 0 & 1 \end{array} \right)$$

1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { 3 }\).

8

1. Find the value of \(a\) for which the system of equations $$\begin{gathered} 3 x + a y = 0 \\ 5 x - y = 0 \\ x + 3 y + 2 z = 0 \end{gathered}$$ does not have a unique solution.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 0 & 0 \\ 5 & - 1 & 0 \\ 1 & 3 & 2 \end{array} \right)$$
2. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
3. Use the characteristic equation of \(\mathbf { A }\) to show that $$( \mathbf { A } + 6 \mathbf { I } ) ^ { 2 } = \mathbf { A } ^ { 4 } ( \mathbf { A } + b \mathbf { I } ) ^ { 2 }$$ where \(b\) is an integer to be determined.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

5 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 18 & 5 & - 11 \\ 8 & 6 & - 4 \\ 32 & 10 & - 20 \end{array} \right)$$

1. Show that the characteristic equation of \(\mathbf { A }\) is \(\lambda ^ { 3 } - 4 \lambda ^ { 2 } - 20 \lambda + 48 = 0\) and hence find the eigenvalues of \(\mathbf { A }\).
2. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).

8 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } a & - 6 a & 2 a + 2 \\ 0 & 1 - a & 0 \\ 0 & 2 - a & - 1 \end{array} \right)$$ where \(a\) is a constant with \(a \neq 0\) and \(a \neq 1\).

1. Show that the equation \(\mathbf { A } \left( \begin{array} { c } x \\ y \\ z \end{array} \right) = \left( \begin{array} { c } 1 \\ 2 \\ 3 \end{array} \right)\) has a unique solution and interpret this situation geometrically.
2. Show that the eigenvalues of \(\mathbf { A }\) are \(a , 1 - a\) and - 1 .
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 4 } = \mathbf { P D P } ^ { - 1 }\).
4. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { 4 }\) in terms of \(\mathbf { A }\) and \(a\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

8

1. Find the set of values of \(a\) for which the system of equations $$\begin{array} { c l } 6 x + a y & = 3 \\ 2 x - y & = 1 \\ x + 5 y + 4 z & = 2 \end{array}$$ has a unique solution.
2. Show that the system of equations in part (a) is consistent for all values of \(a\).
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 6 & 0 & 0 \\ 2 & - 1 & 0 \\ 1 & 5 & 4 \end{array} \right)$$
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( 14 \mathbf { A } + 24 \mathbf { I } ) ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
4. Use the characteristic equation of \(\mathbf { A }\) to show that $$( 14 \mathbf { A } + 24 \mathbf { I } ) ^ { 2 } = \mathbf { A } ^ { 4 } ( \mathbf { A } + b \mathbf { I } ) ^ { 2 }$$ where \(b\) is an integer to be determined.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

6 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } 2 & - 3 & - 7 \\ 0 & 5 & 7 \\ 0 & 0 & - 2 \end{array} \right) .$$

1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
2. Use the characteristic equation of \(\mathbf { A }\) to show that $$\mathbf { A } ^ { 4 } = a \mathbf { A } ^ { 2 } + b \mathbf { I } ,$$ where \(a\) and \(b\) are integers to be determined.

7 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } - 6 & 2 & 13 \\ 0 & - 2 & 5 \\ 0 & 0 & 8 \end{array} \right) .$$

1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { - 1 } = \mathbf { P D P } ^ { - 1 }\).
2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).

8 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 0 & 0 \\ 0 & 7 & 9 \\ 4 & 1 & 7 \end{array} \right)$$

1. Show that the characteristic equation of \(\mathbf { A }\) is \(\lambda ^ { 3 } - 12 \lambda ^ { 2 } + 12 \lambda + 80 = 0\) and find the eigenvalues of A. \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-16_2718_38_106_2010} \includegraphics[max width=\textwidth, alt={}, center]{bc601b16-c106-43a2-a2fc-676b5c836096-17_2723_33_99_22}
2. Use the characteristic equation of \(\mathbf { A }\) to show that $$\mathbf { A } ^ { 4 } = p \mathbf { A } ^ { 2 } + q \mathbf { A } + r \mathbf { I } ,$$ where \(p , q\) and \(r\) are integers to be determined.
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } - 3 \mathbf { I } ) ^ { 4 } = \mathbf { P D P } ^ { - 1 }\) .
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

3 Let \(\mathbf { M } = \left( \begin{array} { r r r } 3 & 5 & 2 \\ 5 & 3 & - 2 \\ 2 & - 2 & - 4 \end{array} \right)\).

1. Show that the characteristic equation for \(\mathbf { M }\) is \(\lambda ^ { 3 } - 2 \lambda ^ { 2 } - 48 \lambda = 0\). You are given that \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\) corresponding to the eigenvalue 0 .
2. Find the other two eigenvalues of \(\mathbf { M }\), and corresponding eigenvectors.
3. Write down a matrix \(\mathbf { P }\), and a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { P } ^ { - 1 } \mathbf { M } ^ { 2 } \mathbf { P } = \mathbf { D }\).
4. Use the Cayley-Hamilton theorem to find integers \(a\) and \(b\) such that \(\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M }\). Section B (18 marks)

3 The matrix \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3 \\ - 2 & - 3 & 6 \\ 2 & 2 & - 4 \end{array} \right)\).

1. Show that the characteristic equation for \(\mathbf { M }\) is \(\lambda ^ { 3 } + 6 \lambda ^ { 2 } - 9 \lambda - 14 = 0\).
2. Show that - 1 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
3. Find an eigenvector corresponding to the eigenvalue - 1 .
4. Verify that \(\left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { r } 0 \\ 3 \\ - 2 \end{array} \right)\) are eigenvectors of \(\mathbf { M }\).
5. Write down a matrix \(\mathbf { P }\), and a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { M } ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
6. Use the Cayley-Hamilton theorem to express \(\mathbf { M } ^ { - 1 }\) in the form \(a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I }\). Section B (18 marks)

11 Find the eigenvalues of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 1 & 4 \\ 1 & 1 & - 1 \\ 2 & 1 & 1 \end{array} \right)$$ and corresponding eigenvectors. The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \mathbf { A } - k \mathbf { I } ,$$ where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix and \(k\) is a real number. Find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { B } ^ { 3 } = \mathbf { P D } \mathbf { P } ^ { - 1 } .$$