Find eigenvectors given eigenvalue

8 The lifetime, in years, of an electrical component is the random variable \(T\), with probability density function f given by $$\mathrm { f } ( t ) = \begin{cases} A \mathrm { e } ^ { - \lambda t } & t \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$ where \(A\) and \(\lambda\) are positive constants.

1. Show that \(A = \lambda\). It is known that out of 100 randomly chosen components, 16 failed within the first year.
2. Find an estimate for the value of \(\lambda\), and hence find an estimate for the median value of \(T\).

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 \(d\).

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.

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 \(l _ { 2 }\) .
   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 \(\mathbf { M }\) ,
   1. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\)
   2. 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] \(\left[ \begin{array} { l l } 4 & 6 \\ 2 & 5 \end{array} \right]\) \(\left[ \begin{array} { l l } 6 & 5 \\ 4 & 2 \end{array} \right]\) \(\left[ \begin{array} { l l } 5 & 2 \\ 6 & 4 \end{array} \right]\) \(\left[ \begin{array} { l l } 2 & 4 \\ 5 & 6 \end{array} \right]\)

The vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) and \(\mathbf{d}\) in \(\mathbb{R}^3\) are given by $$\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 2 \\ 9 \\ 0 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 3 \\ 5 \\ 4 \end{pmatrix} \quad \text{and} \quad \mathbf{d} = \begin{pmatrix} 0 \\ -8 \\ -3 \end{pmatrix}.$$

1. Show that \(\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}\) is a basis for \(\mathbb{R}^3\). [3]
2. Express \(\mathbf{d}\) in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\). [2]

\(\mathbf{M} = \begin{pmatrix} -1 & 2 & -1 \\ 2 & 2 & -2 \\ -1 & -2 & -1 \end{pmatrix}\)

1. Given that 4 is an eigenvalue of M, find a corresponding eigenvector. [3 marks]
2. Given that \(\mathbf{MU} = \mathbf{UD}\), where D is a diagonal matrix, find possible matrices for D and U. [8 marks]

The matrix A is \(\begin{pmatrix} 0.6 & 0.8 \\ 0.8 & -0.6 \end{pmatrix}\)

1. Given that A represents a reflection, write down the eigenvalues of A. [1]
2. Hence find the eigenvectors of A. [3]
3. Write down the equation of the mirror line of the reflection represented by A. [1]

Three matrices, A, B and C, are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ a & -1 \end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix} 2 & -1 \\ 4 & 1 \end{pmatrix}\) and \(\mathbf{C} = \begin{pmatrix} 5 & 0 \\ -2 & 2 \end{pmatrix}\) where \(a\) is a constant.

1. Using A, B and C in that order demonstrate explicitly the associativity property of matrix multiplication. [4]
2. Use A and C to disprove by counterexample the proposition 'Matrix multiplication is commutative'. [2]

For a certain value of \(a\), \(\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix} = 3\begin{pmatrix} x \\ y \end{pmatrix}\)

1. Find