Find P and D for A = PDP⁻¹

8

1. Find the value of \(a\) for which the system of equations $$\begin{array} { r } 13 x + 18 y - 28 z = 0 \\ - 4 x - a y + 8 z = 0 \\ 2 x + 6 y - 5 z = 0 \end{array}$$ does not have a unique solution.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 13 & 18 & - 28 \\ - 4 & - 1 & 8 \\ 2 & 6 & - 5 \end{array} \right)$$
2. Find the eigenvalue of \(\mathbf { A }\) corresponding to the eigenvector \(\left( \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right)\).
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\).
4. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\) in terms of \(\mathbf { A }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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

1. State the eigenvalues of \(\mathbf { P }\).
2. Use the characteristic equation of \(\mathbf { P }\) to find \(\mathbf { P } ^ { - 1 }\).
   The \(3 \times 3\) matrix \(\mathbf { A }\) has distinct eigenvalues \(b , - 1,1\) with corresponding eigenvectors $$\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } 4 \\ - 1 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 2 \\ 1 \\ 2 \end{array} \right)$$ respectively.
3. Find \(\mathbf { A }\) in terms of b.

9 It is given that \(a\) is a positive constant.

1. Show that the system of equations $$\begin{aligned} a x + ( 2 a + 5 ) y + ( a + 1 ) z & = 1 \\ - 4 y & = 2 \\ 3 y - z & = 3 \end{aligned}$$ has a unique solution and interpret this situation geometrically.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } a & 2 a + 5 & a + 1 \\ 0 & - 4 & 0 \\ 0 & 3 & - 1 \end{array} \right)$$
2. Show that the eigenvalues of \(\mathbf { A }\) are \(a , - 1\) and - 4 .
3. Find a matrix \(\mathbf { P }\) such that $$\mathbf { A } = \mathbf { P } \left( \begin{array} { r r r } a & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 4 \end{array} \right) \mathbf { P } ^ { - 1 } .$$
4. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 7 & 3 \\ - 4 & - 1 \end{array} \right)\).

1. Find the eigenvalues, and corresponding eigenvectors, of the matrix \(\mathbf { M }\).
2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }\).
3. Given that \(\mathbf { M } ^ { n } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\), show that \(a = - \frac { 1 } { 2 } + \frac { 3 } { 2 } \times 5 ^ { n }\), and find similar expressions for \(b , c\) and \(d\). Section B (18 marks)

3 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 3 & 0 \\ 3 & - 2 & - 1 \\ 0 & - 1 & 1 \end{array} \right)\).

1. Show that the characteristic equation of \(\mathbf { M }\) is $$\lambda ^ { 3 } - 13 \lambda + 12 = 0 .$$
2. Find the eigenvalues and corresponding eigenvectors of \(\mathbf { M }\).
3. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { M } ^ { n } = \mathbf { P D P } ^ { - 1 } .$$ (You are not required to calculate \(\mathbf { P } ^ { - 1 }\).)

3

1. 1. Find the eigenvalues and corresponding eigenvectors for the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { l l } 6 & - 3 \\ 4 & - 1 \end{array} \right)$$
   2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\).
2. 1. The \(3 \times 3\) matrix \(\mathbf { B }\) has characteristic equation $$\lambda ^ { 3 } - 4 \lambda ^ { 2 } - 3 \lambda - 10 = 0$$ Show that 5 is an eigenvalue of \(\mathbf { B }\). Show that \(\mathbf { B }\) has no other real eigenvalues.
   2. An eigenvector corresponding to the eigenvalue 5 is \(\left( \begin{array} { r } - 2 \\ 1 \\ 4 \end{array} \right)\). Evaluate \(\mathbf { B } \left( \begin{array} { r } - 2 \\ 1 \\ 4 \end{array} \right)\) and \(\mathbf { B } ^ { 2 } \left( \begin{array} { r } 4 \\ - 2 \\ - 8 \end{array} \right)\).
      Solve the equation \(\mathbf { B } \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } - 20 \\ 10 \\ 40 \end{array} \right)\) for \(x , y , z\).
   3. Show that \(\mathbf { B } ^ { 4 } = 19 \mathbf { B } ^ { 2 } + 22 \mathbf { B } + 40 \mathbf { I }\).

8 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 4 & 1 & - 1 \\ - 4 & - 1 & 4 \\ 0 & - 1 & 5 \end{array} \right)$$ Given that one eigenvector of \(\mathbf { A }\) is \(\left( \begin{array} { r } 1 \\ - 2 \\ - 1 \end{array} \right)\), find the corresponding eigenvalue. Given also that another eigenvalue of \(\mathbf { A }\) is 4, find a corresponding eigenvector. Given further that \(\left( \begin{array} { r } 1 \\ - 4 \\ - 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\), with corresponding eigenvalue 1 , find matrices \(\mathbf { P }\) and \(\mathbf { Q }\), together with a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { A } ^ { 5 } = \mathbf { P D Q }\).

10 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 2 & - 3 \\ 2 & 2 & 3 \\ - 3 & 3 & 3 \end{array} \right)$$ The matrix \(\mathbf { A }\) has an eigenvector \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)\). Find the corresponding eigenvalue. The matrix \(\mathbf { A }\) also has eigenvalues 4 and 6. Find corresponding eigenvectors. Hence find a matrix \(\mathbf { P }\) such that \(\mathbf { A } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\), where \(\mathbf { D }\) is a diagonal matrix which is to be determined. The matrix \(\mathbf { B }\) is such that \(\mathbf { B } = \mathbf { Q A Q } ^ { - 1 }\), where $$\mathbf { Q } = \left( \begin{array} { r r r } 4 & 11 & 5 \\ 1 & 4 & 2 \\ 1 & 2 & 1 \end{array} \right)$$ By using the expression \(\mathbf { P D P } ^ { - 1 }\) for \(\mathbf { A }\), find the set of eigenvalues and a corresponding set of eigenvectors for \(\mathbf { B }\).
[0pt] [Question 11 is printed on the next page.]

10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 1 & - 1 \\ 0 & - 1 & 2 \\ 0 & 0 & 1 \end{array} \right)$$ and find corresponding eigenvectors. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\), and hence find the matrix \(\mathbf { A } ^ { n }\), where \(n\) is a positive integer.
[0pt] [Question 11 is printed on the next page.]

10 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { l l l } 6 & - 8 & 7 \\ 7 & - 9 & 7 \\ 6 & - 6 & 5 \end{array} \right)$$

1. Given that \(\left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\), find the corresponding eigenvalue.
2. Given also that - 1 is an eigenvalue of \(\mathbf { A }\), find a corresponding eigenvector.
3. It is given that the determinant of \(\mathbf { A }\) is equal to the product of the eigenvalues of \(\mathbf { A }\). Use this result to find the third eigenvalue of \(\mathbf { A }\), and find also a corresponding eigenvector.
4. Write down matrices \(\mathbf { P }\) and \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\), where \(\mathbf { D }\) is a diagonal matrix, and hence find the matrix \(\mathbf { A } ^ { n }\) in terms of \(n\), where \(n\) is a positive integer.

It is given that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A }\), with corresponding eigenvalue \(\lambda\).

1. Write down another eigenvector of \(\mathbf { A }\) corresponding to \(\lambda\).
2. Write down an eigenvector and corresponding eigenvalue of \(\mathbf { A } ^ { n }\), where \(n\) is a positive integer.
   Let \(\mathbf { A } = \left( \begin{array} { l l l } 3 & 0 & 0 \\ 2 & 7 & 0 \\ 4 & 8 & 1 \end{array} \right)\).
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
4. Determine the set of values of the real constant \(k\) such that $$\sum _ { n = 1 } ^ { \infty } k ^ { n } \left( \mathbf { A } ^ { n } - k \mathbf { A } ^ { n + 1 } \right) = k \mathbf { A } .$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 4 & - 16 \\ 0 & 2 & 3 \\ 0 & 0 & 3 \end{array} \right)$$ Find corresponding eigenvectors. Let \(n\) be a positive integer. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { A } ^ { n } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$$ Find \(\mathbf { P } ^ { - 1 }\) and \(\mathbf { A } ^ { n }\). Hence find \(\lim _ { n \rightarrow \infty } \left( 3 ^ { - n } \mathbf { A } ^ { n } \right)\).

8 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left( \begin{array} { c c c } 2 & m & 1 \\ 0 & m & 7 \\ 0 & 0 & 1 \end{array} \right) ,$$ where \(m \neq 0,1,2\).

1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } = \mathbf { P D P } ^ { - 1 }\).
2. Find \(\mathbf { M } ^ { 7 } \mathbf { P }\).

10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 4 & - 16 \\ 0 & 2 & 3 \\ 0 & 0 & 3 \end{array} \right)$$ Find corresponding eigenvectors. Let \(n\) be a positive integer. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { A } ^ { n } = \mathbf { P D } \mathbf { P } ^ { - 1 }$$ Find \(\mathbf { P } ^ { - 1 }\) and \(\mathbf { A } ^ { n }\). Hence find \(\lim _ { n \rightarrow \infty } \left( 3 ^ { - n } \mathbf { A } ^ { n } \right)\).

4 The matrix \(\mathbf { P }\) is given by \(\mathbf { P } = \left( \begin{array} { r r r } 1 & 7 & 8 \\ - 6 & 12 & 12 \\ - 2 & 4 & 8 \end{array} \right)\).

1. Show that the characteristic equation of \(\mathbf { P }\) is \(- \lambda ^ { 3 } + 21 \lambda ^ { 2 } - 126 \lambda + 216 = 0\). You are given that the roots of this equation are 3,6 and 12 .
2. 1. Verify that \(\left( \begin{array} { r } 1 \\ - 2 \\ 2 \end{array} \right)\) is an eigenvector of \(\mathbf { P }\), stating its associated eigenvalue.
   2. The vector \(\left( \begin{array} { l } x \\ y \\ z \end{array} \right)\) is an eigenvector of \(\mathbf { P }\) with eigenvalue 6. Given that \(z = 5\), find \(x\) and \(y\). You are given that \(\mathbf { P }\) can be expressed in the form \(\mathbf { E D E } ^ { - 1 }\), where \(\mathbf { E } = \left( \begin{array} { r r r } 3 & 2 & 1 \\ 1 & 2 & - 2 \\ 1 & 1 & 2 \end{array} \right)\) and \(\mathbf { D }\) is a diagonal matrix. The characteristic equation of \(\mathbf { E }\) is \(- \lambda ^ { 3 } + 7 \lambda ^ { 2 } - 15 \lambda + 9 = 0\).
3. 1. Use the Cayley-Hamilton theorem to express \(\mathbf { E } ^ { - 1 }\) in terms of positive powers of \(\mathbf { E }\).
   2. Hence find \(\mathbf { E } ^ { - 1 }\).
   3. By identifying the matrix \(\mathbf { D }\) and using \(\mathbf { P } = \mathbf { E D E } ^ { - 1 }\), determine \(\mathbf { P } ^ { 4 }\).

4. $$\mathbf { A } = \left( \begin{array} { r r } 1 & 1 \\ - 2 & 4 \end{array} \right)$$ Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { D } = \mathbf { P } ^ { - 1 } \mathbf { A P }\)

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

$$\mathbf { M } = \left( \begin{array} { r r } 4 & 2 \\ 3 & - 1 \end{array} \right)$$ Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }$$

1. The matrix \(\mathbf { A }\) is given by

$$\mathbf { A } = \left( \begin{array} { r r r } 6 & - 2 & 2 \\ - 2 & 3 & - 1 \\ 2 & - 1 & 3 \end{array} \right)$$

1. Show that 2 is a repeated eigenvalue of \(\mathbf { A }\) and find the other eigenvalue.
2. Hence find three non-parallel eigenvectors of \(\mathbf { A }\).
3. Find a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P }\) is a diagonal matrix.

8. $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 2 & 5 \\ 0 & 3 & p \\ - 6 & 6 & - 4 \end{array} \right) \quad \text { where } p \text { is a constant }$$ Given that \(\left( \begin{array} { r } 2 \\ 1 \\ - 2 \end{array} \right)\) is an eigenvector for \(\mathbf { A }\)

1. 1. determine the eigenvalue corresponding to this eigenvector
   2. hence show that \(p = 2\)
   3. determine the remaining eigenvalues and corresponding eigenvectors of \(\mathbf { A }\)
2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\)
3. 1. Solve the differential equation \(\dot { u } = k u\), where \(k\) is a constant. With respect to a fixed origin \(O\), the velocity of a particle moving through space is modelled by $$\left( \begin{array} { c } \dot { x } \\ \dot { y } \\ \dot { z } \end{array} \right) = \mathbf { A } \left( \begin{array} { l } x \\ y \\ z \end{array} \right)$$ By considering \(\left( \begin{array} { c } u \\ v \\ w \end{array} \right) = \mathbf { P } ^ { - 1 } \left( \begin{array} { c } x \\ y \\ z \end{array} \right)\) so that \(\left( \begin{array} { c } \dot { u } \\ \dot { v } \\ \dot { w } \end{array} \right) = \mathbf { P } ^ { - 1 } \left( \begin{array} { c } \dot { x } \\ \dot { y } \\ \dot { z } \end{array} \right)\)
   2. determine a general solution for the displacement of the particle.

4. $$\mathbf { A } = \left( \begin{array} { r r r } 4 & 2 & 0 \\ 2 & p & - 2 \\ 0 & - 2 & 2 \end{array} \right) \quad \text { where } p \text { is a constant }$$ Given that \(\left( \begin{array} { r } 2 \\ - 1 \\ 2 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\),

1. determine the eigenvalue corresponding to this eigenvector.
2. Hence show that \(p = 3\)
3. Determine
   1. the remaining eigenvalues of \(\mathbf { A }\),
   2. corresponding eigenvectors for these eigenvalues.
4. Hence determine a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { \mathrm { T } }\)

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

$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 0 \\ 1 & 2 & 0 \\ - 1 & 0 & 4 \end{array} \right)$$

1. Show that 4 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
2. For each of the eigenvalues find a corresponding eigenvector.
3. Find a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P }\) is a diagonal matrix.

11

1. Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf { M } = \left[ \begin{array} { c c } \frac { 5 } { 2 } & - \frac { 3 } { 2 } \\ - \frac { 3 } { 2 } & \frac { 13 } { 2 } \end{array} \right]$$ 11
2. (i) Describe how the directions of the invariant lines of the transformation represented by \(\mathbf { M }\) are related to each other. Fully justify your answer.
   [0pt] [2 marks]
   11 (b) (ii) Describe fully the transformation represented by \(\mathbf { M }\)

6 The matrix A, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0 \\ 10 & - 7 & 10 \\ 7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. It is given that \(\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\). Find the corresponding eigenvalue. Find a diagonal matrix \(\mathbf { D }\) and matrices \(\mathbf { P }\) and \(\mathbf { P } ^ { - 1 }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\).

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]

The matrix \(\mathbf{A}\), where $$\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 10 & -7 & 10 \\ 7 & -5 & 8 \end{pmatrix},$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. [3] It is given that \(\begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix}\) is an eigenvector of \(\mathbf{A}\). Find the corresponding eigenvalue. [2] Find a diagonal matrix \(\mathbf{D}\) and matrices \(\mathbf{P}\) and \(\mathbf{P}^{-1}\) such that \(\mathbf{P}^{-1}\mathbf{A}\mathbf{P} = \mathbf{D}\). [5]