Invariant lines and eigenvalues and vectors

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 \(\mathbf { D } = \mathbf { P } ^ { - 1 } \mathbf { A P }\)

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)

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.

22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r }

22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r }

4 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } k & 0 & 2 \\ 0 & - 1 & - 1 \\ 1 & 1 & - k \end{array} \right)$$ where \(k\) is a real constant.

1. Show that \(\mathbf { A }\) is non-singular.
   The matrices \(\mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { B } = \left( \begin{array} { r r } 0 & - 3 \\ - 1 & 3 \\ 0 & 0 \end{array} \right) \text { and } \mathbf { C } = \left( \begin{array} { r r r } - 3 & - 1 & 1 \\ 1 & 1 & 2 \end{array} \right)$$ It is given that \(\mathbf { C A B } = \left( \begin{array} { l l } - 2 & - \frac { 3 } { 2 } \\ - 1 & - \frac { 3 } { 2 } \end{array} \right)\).
2. Find the value of \(k\).
3. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\).

Find two invariant points under the transformation given by \(\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}\) [2 marks]

The matrix \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\) represents a transformation. Which one of the points below is an invariant point under this transformation? Circle your answer. [1 mark] \((1, 1) \quad (0, 2) \quad (3, 0) \quad (2, 1)\)

3 A surface, \(S\), is defined by \(g ( x , y , z ) = 0\) where \(g ( x , y , z ) = 2 x ^ { 3 } - x ^ { 2 } y + 2 x y ^ { 2 } + 27 z\). The normal to \(S\) at the point \(\left( 1,1 , - \frac { 1 } { 9 } \right)\) and the tangent plane to \(S\) at the point \(( 3,3 , - 3 )\) intersect at \(P\). Determine the position vector of P .

7 The square matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\) and state the corresponding eigenvalue. Find the eigenvalues of the matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { l l l }

1. The matrix \(\mathbf{S} = \begin{pmatrix} -1 & 2 \\ -3 & 4 \end{pmatrix}\) represents a transformation.
   1. Show that the point \((1, 1)\) is invariant under this transformation. [1]
   2. Calculate \(\mathbf{S}^{-1}\). [2]
   3. Verify that \((1, 1)\) is also invariant under the transformation represented by \(\mathbf{S}^{-1}\). [1]
2. Part (i) may be generalised as follows. If \((x, y)\) is an invariant point under a transformation represented by the non-singular matrix \(\mathbf{T}\), it is also invariant under the transformation represented by \(\mathbf{T}^{-1}\). Starting with \(\mathbf{T}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}\), or otherwise, prove this result. [2]

7

1. It is given that \(\lambda\) is an eigenvalue of the non-singular square matrix \(\mathbf { A }\), with corresponding eigenvector \(\mathbf { e }\). Show that \(\lambda ^ { - 1 }\) is an eigenvalue of \(\mathbf { A } ^ { - 1 }\) for which \(\mathbf { e }\) is a corresponding eigenvector.
   The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 3 \\ 15 & - 4 & 3 \\ 3 & 0 & 2 \end{array} \right)$$
2. Given that - 1 is an eigenvalue of \(\mathbf { A }\), find a corresponding eigenvector.
3. It is also given that \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)\) are eigenvectors of \(\mathbf { A }\). Find the corresponding eigenvalues.
4. Hence find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { - 1 } = \mathbf { P D P } ^ { - 1 }\).
5. Use the characteristic equation of \(\mathbf { A }\) to show that \(\mathbf { A } ^ { - 1 } = p \mathbf { A } ^ { 2 } + q l\), where \(p\) and \(q\) are rational numbers to be determined.

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.

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] \(\left[ \begin{array} { l l } 4 & 6 \\ 2 & 5 \end{array} \right]\) \(\left[ \begin{array} { l l } 6 & 5 \\ 4 & 2 \end{array} \right]\) \(\left[ \begin{array} { l l } 5 & 2 \\ 6 & 4 \end{array} \right]\) \(\left[ \begin{array} { l l } 2 & 4 \\ 5 & 6 \end{array} \right]\)

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.

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

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.

1 Given that 5 is an eigenvalue of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 3 & 0 \\ 1 & 2 & 1 \\ - 1 & 3 & 4 \end{array} \right)$$ find a corresponding eigenvector. Hence find an eigenvalue and a corresponding eigenvector of the matrix \(\mathbf { A } + \mathbf { A } ^ { 2 }\).

1 Given that 5 is an eigenvalue of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 3 & 0 \\ 1 & 2 & 1 \\ - 1 & 3 & 4 \end{array} \right)$$ find a corresponding eigenvector. Hence find an eigenvalue and a corresponding eigenvector of the matrix \(\mathbf { A } + \mathbf { A } ^ { 2 }\).

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

1. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\), with corresponding eigenvalue \(\lambda ^ { 2 }\).
   The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left( \begin{array} { c c c } n & 1 & 3 \\ 0 & 2 n & 0 \\ 0 & 0 & 3 n \end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + n \mathbf { I } ) ^ { 2 }$$ where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix and \(n\) is a non-zero integer.
2. Find, in terms of \(n\), a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { B } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).

1 & - 3 & - 2 \end{array} \right) \left( \begin{array} { l } x
y
z \end{array} \right) = \left( \begin{array} { c } p

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.

The matrix \(\mathbf{C}\) is defined by $$\mathbf{C} = \begin{bmatrix} 3 & 2 \\ -4 & 5 \end{bmatrix}$$ Prove that the transformation represented by \(\mathbf{C}\) has no invariant lines of the form \(y = kx\) [4 marks]

1. Given that

$$A = \left( \begin{array} { l l } 3 & 1 \\ 6 & 4 \end{array} \right)$$

1. find the characteristic equation of the matrix \(\mathbf { A }\).
2. Hence show that \(\mathbf { A } ^ { 3 } = 43 \mathbf { A } - 42 \mathbf { I }\).

2. $$\mathbf { A } = \left( \begin{array} { l l } 3 & 2 \\ 2 & 6 \end{array} \right)$$

1. Find the eigenvalues and corresponding normalised eigenvectors of the matrix \(\mathbf { A }\).
2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { \mathrm { T } } \mathbf { A P } = \mathbf { D }\).

Find the eigenvalues of the matrix \(\begin{pmatrix} 7 & 6 \\ 6 & 2 \end{pmatrix}\) (Total 4 marks)

4 In this question you must show detailed reasoning. \(\mathbf { M }\) is the matrix \(\left( \begin{array} { l l } 1 & 6 \\ 0 & 2 \end{array} \right)\).
Prove that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & 3 \left( 2 ^ { n + 1 } - 2 \right) \\ 0 & 2 ^ { n } \end{array} \right)\), for any positive integer \(n\).

1. A linear transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix

$$\mathbf { M } = \left( \begin{array} { c c } 5 & 1 \\ k & - 3 \end{array} \right)$$ where \(k\) is a constant.
Given that matrix \(\mathbf { M }\) has a repeated eigenvalue,

1. determine
   1. the value of \(k\)
   2. the eigenvalue.
2. Hence determine a Cartesian equation of the invariant line under \(T\).

4. $$\mathbf { M } = \left( \begin{array} { l l l } 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{array} \right)$$

1. Show that 6 is an eigenvalue of the matrix \(\mathbf { M }\) and find the other two eigenvalues of \(\mathbf { M }\).
2. Find a normalised eigenvector corresponding to the eigenvalue 6

9 A linear transformation of the plane is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 1 & - 2 \\ \lambda & 3 \end{array} \right)\), where \(\lambda\) is a
constant. constant.

1. Find the set of values of \(\lambda\) for which the linear transformation has no invariant lines through the origin.
2. Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines.

7 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 0.6 & 2.4 \\ - 0.8 & 1.8 \end{array} \right)\).

1. Find \(\operatorname { det } \mathbf { A }\). The matrix A represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.
2. By considering the determinants of these transformations, determine the scale factor of the stretch.
3. Explain whether the stretch is parallel to the \(x\)-axis or the \(y\)-axis, justifying your answer.
4. Find the angle of rotation.

3 Find a matrix \(\mathbf { A }\) whose eigenvalues are \(- 1,1,2\) and for which corresponding eigenvectors are $$\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 0 \\ 1 \\ 1 \end{array} \right) ,$$ respectively.

The linear transformation \(\mathrm{T} : \mathbb{R}^4 \to \mathbb{R}^4\) is represented by the matrix \(\mathbf{M}\), where $$\mathbf{M} = \begin{pmatrix} 3 & 2 & 0 & 1 \\ 6 & 5 & -1 & 3 \\ 9 & 8 & -2 & 5 \\ -3 & -2 & 0 & -1 \end{pmatrix}.$$

1. Find the rank of \(\mathbf{M}\). [3]

Let \(K\) be the null space of \(\mathrm{T}\).

1. Find a basis for \(K\). [3]
2. Find the general solution of $$\mathbf{M}\mathbf{x} = \begin{pmatrix} 2 \\ 5 \\ 8 \\ -2 \end{pmatrix}.$$ [3]