Find eigenvectors given eigenvalue

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7. \(\quad \mathbf { A } = \left( \begin{array} { c c c } 2 & \mathrm { k } & 0 \\ 1 & 1 & 0 \\ 0 & - 2 & 1 \end{array} \right)\), where k is a constant. Given that \(\left( \begin{array} { c } 9 \\ 3 \\ - 2 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\),

1. show that \(\mathrm { k } = 6\),
2. find the eigenvalues of \(\mathbf { A }\). A transformation \(\mathrm { T } : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { A }\).
   The point P has coordinates \(( \mathrm { t } - 2 , \mathrm { t } , 2 \mathrm { t } )\) where t is a parameter.
3. Show that, for any value of \(t\), the transformation \(T\) maps \(P\) onto a point on the line with equation \(x - 4 y - 4 = 0\) (5)

6. \(\mathbf { M } = \left( \begin{array} { c c c } 1 & 0 & 3 \\ 0 & - 2 & 1 \\ k & 0 & 1 \end{array} \right)\), where \(k\) is a constant. Given that \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\),

1. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\),
2. show that \(k = 3\),
3. show that \(\mathbf { M }\) has exactly two eigenvalues. A transformation \(T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by \(\mathbf { M }\).
   The transformation \(T\) maps the line \(l _ { 1 }\), with cartesian equations \(\frac { x - 2 } { 1 } = \frac { y } { - 3 } = \frac { z + 1 } { 4 }\), onto the line \(l _ { 2 }\).
4. Taking \(k = 3\), find cartesian equations of \(l _ { 2 }\).

6. It is given that \(\left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { l l l } 4 & 2 & 3 \\ 2 & b & 0 \\ a & 1 & 8 \end{array} \right)$$ and \(a\) and \(b\) are constants.

1. Find the eigenvalue of \(\mathbf { A }\) corresponding to the eigenvector \(\left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\).
2. Find the values of \(a\) and \(b\).
3. Find the other eigenvalues of \(\mathbf { A }\).

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

$$\mathbf { M } = \left( \begin{array} { r r r } 1 & 1 & a \\ 2 & b & c \\ - 1 & 0 & 1 \end{array} \right) , \text { where } a , b \text { and } c \text { are constants. }$$

1. Given that \(\mathbf { j } + \mathbf { k }\) and \(\mathbf { i } - \mathbf { k }\) are two of the eigenvectors of \(\mathbf { M }\), find
   1. the values of \(a , b\) and \(c\),
   2. the eigenvalues which correspond to the two given eigenvectors.
2. The matrix \(\mathbf { P }\) is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & 1 & 0 \\ 2 & 1 & d \\ - 1 & 0 & 1 \end{array} \right) \text {, where } d \text { is constant, } d \neq - 1$$ Find
   1. the determinant of \(\mathbf { P }\) in terms of \(d\),
   2. the matrix \(\mathbf { P } ^ { - 1 }\) in terms of \(d\).

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   ACHIEVEMENT

1 It is given that $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & - 2 \\ 0 & 2 & 1 \\ 0 & 0 & - 3 \end{array} \right)$$ Write down the eigenvalues of \(\mathbf { A }\) and find corresponding eigenvectors.

Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 1 & 2 \\ 0 & 2 & 2 \\ - 1 & 1 & 3 \end{array} \right)$$ The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is defined by \(\mathbf { x } \mapsto \mathbf { A x }\). Let \(\mathbf { e } , \mathbf { f }\) be two linearly independent eigenvectors of \(\mathbf { A }\), with corresponding eigenvalues \(\lambda\) and \(\mu\) respectively, and let \(\Pi\) be the plane, through the origin, containing \(\mathbf { e }\) and \(\mathbf { f }\). By considering the parametric equation of \(\Pi\), show that all points of \(\Pi\) are mapped by T onto points of \(\Pi\). Find cartesian equations of three planes, each with the property that all points of the plane are mapped by T onto points of the same plane.

9
0 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l } 3
3
4 \end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r } 0
- 8
3 \end{array} \right) .$$

1. Show that \(\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}\) is a basis for \(\mathbb { R } ^ { 3 }\).
2. Express \(\mathbf { d }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\).\\ 2 The roots of the equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ are \(\alpha , 2 \alpha , 4 \alpha\), where \(p , q , r\) and \(\alpha\) are non-zero real constants.
3. Show that $$2 p \alpha + q = 0$$
4. Show that $$p ^ { 3 } r - q ^ { 3 } = 0$$ 3 The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } < 3\) and, for \(n \geqslant 1\), $$u _ { n + 1 } = \frac { 4 u _ { n } + 9 } { u _ { n } + 4 }$$
5. By considering \(3 - u _ { n + 1 }\), or otherwise, prove by mathematical induction that \(u _ { n } < 3\) for all positive integers \(n\).
6. Show that \(u _ { n + 1 } > u _ { n }\) for \(n \geqslant 1\).\\ 4 A curve is defined parametrically by $$x = t - \frac { 1 } { 2 } \sin 2 t \quad \text { and } \quad y = \sin ^ { 2 } t$$ The arc of the curve joining the point where \(t = 0\) to the point where \(t = \pi\) is rotated through one complete revolution about the \(x\)-axis. The area of the surface generated is denoted by \(S\).
7. Show that $$S = a \pi \int _ { 0 } ^ { \pi } \sin ^ { 3 } t \mathrm {~d} t$$ where the constant \(a\) is to be found.
8. Using the result \(\sin 3 t = 3 \sin t - 4 \sin ^ { 3 } t\), find the exact value of \(S\).\\ 5 It is given that \(\lambda\) is an eigenvalue of the matrix \(\mathbf { A }\) with \(\mathbf { e }\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the matrix \(\mathbf { B }\) for which \(\mathbf { e }\) is also a corresponding eigenvector.
9. Show that \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf { A } + \mathbf { B }\) with \(\mathbf { e }\) as a corresponding eigenvector.\\ The matrix \(\mathbf { A }\), given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 1
   - 1 & 2 & 3
   1 & 0 & 2 \end{array} \right)$$ has \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) as eigenvectors.
10. Find the corresponding eigenvalues.\\ The matrix \(\mathbf { B }\) has eigenvalues 4, 5 and 1 with corresponding eigenvectors \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) respectively.
11. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 3 } = \mathbf { P D P } ^ { - 1 }\).\\ 6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } + a x - 1 } { x + 1 }$$ where \(a\) is constant and \(a > 1\).
12. Find the equations of the asymptotes of \(C\).
13. Show that \(C\) intersects the \(x\)-axis twice.
14. Justifying your answer, find the number of stationary points on \(C\).
15. Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis. 7
16. Use de Moivre's theorem to show that $$\sin 8 \theta = 8 \sin \theta \cos \theta \left( 1 - 10 \sin ^ { 2 } \theta + 24 \sin ^ { 4 } \theta - 16 \sin ^ { 6 } \theta \right) .$$
17. Use the equation \(\frac { \sin 8 \theta } { \sin 2 \theta } = 0\) to find the roots of $$16 x ^ { 6 } - 24 x ^ { 4 } + 10 x ^ { 2 } - 1 = 0$$ in the form \(\sin k \pi\), where \(k\) is rational.\\ 8 The plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 5
    1
    0 \end{array} \right) + s \left( \begin{array} { r } - 4
    1
    3 \end{array} \right) + t \left( \begin{array} { l } 0
    1
    2 \end{array} \right)$$
18. Find a cartesian equation of \(\Pi _ { 1 }\).\\ The plane \(\Pi _ { 2 }\) has equation \(3 x + y - z = 3\).
19. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in degrees.
20. Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). [5]\\ 9 The curve \(C\) has polar equation $$r = 5 \sqrt { } ( \cot \theta ) ,$$ where \(0.01 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
21. Find the area of the finite region bounded by \(C\) and the line \(\theta = 0.01\), showing full working. Give your answer correct to 1 decimal place.
    Let \(P\) be the point on \(C\) where \(\theta = 0.01\).
22. Find the distance of \(P\) from the initial line, giving your answer correct to 1 decimal place.
23. Find the maximum distance of \(C\) from the initial line.
24. Sketch \(C\).

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1. Given that 4 is an eigenvalue of \(\mathbf { M }\), find a corresponding eigenvector.
   [0pt] [3 marks] 12
2. Given that \(\mathbf { M U } = \mathbf { U D }\), where \(\mathbf { D }\) is a diagonal matrix, find possible matrices for \(\mathbf { D }\) and \(\mathbf { U }\). [8 marks] \(13 \quad \mathbf { S }\) is a singular matrix such that $$\operatorname { det } \mathbf { S } = \left| \begin{array} { c c c } a & a & x \\ x - b & a - b & x + 1 \\ x ^ { 2 } & a ^ { 2 } & a x \end{array} \right|$$ Express the possible values of \(x\) in terms of \(a\) and \(b\).
   [0pt] [7 marks]

1 The matrix \(\mathbf { A }\) is \(\left( \begin{array} { r r } 0.6 & 0.8 \\ 0.8 & - 0.6 \end{array} \right)\).

1. Given that \(\mathbf { A }\) represents a reflection, write down the eigenvalues of \(\mathbf { A }\).
2. Hence find the eigenvectors of \(\mathbf { A }\).
3. Write down the equation of the mirror line of the reflection represented by \(\mathbf { A }\).

5 In this question you must show detailed reasoning. You are given that the matrix \(\mathbf { M } = \left( \begin{array} { c c c } \frac { 1 } { 2 } & - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { 2 } \\ \frac { 1 } { \sqrt { 2 } } & 0 & - \frac { 1 } { \sqrt { 2 } } \\ \frac { 1 } { 2 } & \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { 2 } \end{array} \right)\) represents a rotation in 3-D space.

1. Explain why it follows that \(\mathbf { M }\) has 1 as an eigenvalue.
2. Find a vector equation for the axis of the rotation.
3. Show that the characteristic equation of \(\mathbf { M }\) can be written as $$\lambda ^ { 3 } - \lambda ^ { 2 } + \lambda - 1 = 0 .$$
4. Find the smallest positive integer \(n\) such that \(\mathbf { M } ^ { n } = \mathbf { I }\).
5. Find the magnitude of the angle of the rotation which \(\mathbf { M }\) represents. Give your reasoning. {www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
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6 Three matrices, \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\), are given by \(\mathbf { A } = \left( \begin{array} { c c } 1 & 2 \\ a & - 1 \end{array} \right) , \mathbf { B } = \left( \begin{array} { c c } 2 & - 1 \\ 4 & 1 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { c c } 5 & 0 \\ - 2 & 2 \end{array} \right)\) where \(a\) is a

1. Using \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) in that order demonstrate explicitly the associativity property of matrix multiplication.
2. Use \(\mathbf { A }\) and \(\mathbf { C }\) to disprove by counterexample the proposition 'Matrix multiplication is commutative'. For a certain value of \(a , \mathbf { A } \binom { x } { y } = 3 \binom { x } { y }\).
3. Find
   • \(y\) in terms of \(x\),
   • the value of \(a\). \(7 C\) is the locus of numbers, \(z\), for which \(\operatorname { Im } \left( \frac { z + 7 \mathrm { i } } { z - 24 } \right) = \frac { 1 } { 4 }\).
     By writing \(z = x + \mathrm { i } y\) give a complete description of the shape of \(C\) on an Argand diagram.

6. \(\mathbf { M } = \left( \begin{array} { c c c } 1 & 0 & 3 \\ 0 & - 2 & 1 \\ k & 0 & 1 \end{array} \right)\), where \(k\) is a constant．
Given that \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\) ，
（a）find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\) ，
（b）show that \(k = 3\) ，
（c）show that \(\mathbf { M }\) has exactly two eigenvalues． A transformation \(T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by \(\mathbf { M }\) ．
The transformation \(T\) maps the line \(l _ { 1 }\) ，with cartesian equations \(\frac { x - 2 } { 1 } = \frac { y } { - 3 } = \frac { z + 1 } { 4 }\) ，onto the line \(l _ { 2 }\) ．
6. \(\mathbf { M } = \left( \begin{array} { c c c } 0 & - 2 & 1 \\ k & 0 & 1 \end{array} \right)\), where \(k\) is a constant． Given that \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\) ，
（a）find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\) （d）Taking \(k = 3\) ，find cartesian equations of \(l _ { 2 }\) ．

4 The point \(( 2 , - 1 )\) is invariant under the transformation represented by the matrix \(\mathbf { N }\) Which of the following matrices could be \(\mathbf { N }\) ? Circle your answer.
[0pt] [1 mark] \(\left[ \begin{array} { l l } 4 & 6 \\ 2 & 5 \end{array} \right]\) \(\left[ \begin{array} { l l } 6 & 5 \\ 4 & 2 \end{array} \right]\) \(\left[ \begin{array} { l l } 5 & 2 \\ 6 & 4 \end{array} \right]\) \(\left[ \begin{array} { l l } 2 & 4 \\ 5 & 6 \end{array} \right]\)