Prove eigenvalue/eigenvector properties

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.

5

1. The matrix \(\mathbf { S } = \left( \begin{array} { l l } - 1 & 2 \\ - 3 & 4 \end{array} \right)\) represents a transformation.
   (A) Show that the point \(( 1,1 )\) is invariant under this transformation.
   (B) Calculate \(\mathbf { S } ^ { - 1 }\).
   (C) Verify that \(( 1,1 )\) is also invariant under the transformation represented by \(\mathbf { S } ^ { - 1 }\).
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 \(\mathbf { T } \binom { x } { y } = \binom { x } { y }\), or otherwise, prove this result.

3 Show that if \(\lambda\) is an eigenvalue of the square matrix \(\mathbf { A }\) with \(\mathbf { e }\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the square matrix \(\mathbf { B }\) for which \(\mathbf { e }\) is also a corresponding eigenvector, then \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf { A } + \mathbf { B }\) with \(\mathbf { e }\) as a corresponding eigenvector. The matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 1 & 0 \\ - 4 & - 6 & - 6 \\ 5 & 11 & 10 \end{array} \right)$$ has \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)\) as an eigenvector. Find the corresponding eigenvalue. The other two eigenvalues of \(\mathbf { A }\) are 1 and 2, with corresponding eigenvectors \(\left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\) and \(\left( \begin{array} { r } 1 \\ 1 \\ - 2 \end{array} \right)\) respectively. The matrix \(\mathbf { B }\) has eigenvalues \(2,3,1\) with corresponding eigenvectors \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\), \(\left( \begin{array} { r } 1 \\ 1 \\ - 2 \end{array} \right)\) respectively. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 4 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
[0pt] [You are not required to evaluate \(\mathbf { P } ^ { - 1 }\).]

5 The matrix \(\mathbf { A }\) has an eigenvalue \(\lambda\) with corresponding eigenvector \(\mathbf { e }\). Prove that the matrix \(( \mathbf { A } + k \mathbf { I } )\), where \(k\) is a real constant and \(\mathbf { I }\) is the identity matrix, has an eigenvalue ( \(\lambda + k\) ) with corresponding eigenvector \(\mathbf { e }\). The matrix \(\mathbf { B }\) is given by $$\mathbf { B } = \left( \begin{array} { r r r } 2 & 2 & - 3 \\ 2 & 2 & 3 \\ - 3 & 3 & 3 \end{array} \right) .$$ Two of the eigenvalues of \(\mathbf { B }\) are - 3 and 4 . Find corresponding eigenvectors. Given that \(\left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\) is an eigenvector of \(\mathbf { B }\), find the corresponding eigenvalue. Hence find the eigenvalues of \(\mathbf { C }\), where $$\mathbf { C } = \left( \begin{array} { r r r } - 1 & 2 & - 3 \\ 2 & - 1 & 3 \\ - 3 & 3 & 0 \end{array} \right) ,$$ and state corresponding eigenvectors.

9 The square matrix \(\mathbf { A }\) has an eigenvalue \(\lambda\) with corresponding eigenvector \(\mathbf { e }\). The non-singular matrix \(\mathbf { M }\) is of the same order as \(\mathbf { A }\). Show that \(\mathbf { M e }\) is an eigenvector of the matrix \(\mathbf { B }\), where \(\mathbf { B } = \mathbf { M } \mathbf { A } \mathbf { M } ^ { - 1 }\), and that \(\lambda\) is the corresponding eigenvalue. Let $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 2 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{array} \right)$$ Write down the eigenvalues of \(\mathbf { A }\) and obtain corresponding eigenvectors. Given that $$\mathbf { M } = \left( \begin{array} { l l l } 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$$ find the eigenvalues and corresponding eigenvectors of \(\mathbf { B }\).

8 It is given that \(\lambda\) is an eigenvalue of the non-singular square matrix \(\mathbf { A }\), with corresponding eigenvector
e. Show that \(\lambda ^ { - 1 }\) is an eigenvalue of \(\mathbf { A } ^ { - 1 }\) for which \(\mathbf { e }\) is a corresponding eigenvector. Deduce that \(\lambda + \lambda ^ { - 1 }\) is an eigenvalue of \(\mathbf { A } + \mathbf { A } ^ { - 1 }\). It is given that 1 is an eigenvalue of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 1 \\ - 1 & 2 & 3 \\ 1 & 0 & 2 \end{array} \right)$$ Find a corresponding eigenvector. 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 the matrix \(\mathbf { A }\). Find the corresponding eigenvalues.
Hence find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\left( \mathbf { A } + \mathbf { A } ^ { - 1 } \right) ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$$

The vector \(\mathbf { e }\) is an eigenvector of the square matrix \(\mathbf { G }\). Show that

1. \(\mathbf { e }\) is an eigenvector of \(\mathbf { G } + k \mathbf { I }\), where \(k\) is a scalar and \(\mathbf { I }\) is an identity matrix,
2. \(\mathbf { e }\) is an cigenvector of \(\mathbf { G } ^ { 2 }\). Find the eigenvalues, and corresponding eigenvectors, of the matrices \(\mathbf { A }\) and \(\mathbf { B } ^ { 2 }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 3 & 0 \\ 1 & 0 & 1 \\ - 1 & 3 & 2 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r r } - 5 & - 3 & 0 \\ 1 & - 8 & 1 \\ - 1 & 3 & - 6 \end{array} \right)$$

3 Three \(n \times 1\) column vectors are denoted by \(\mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \mathbf { x } _ { 3 }\), and \(\mathbf { M }\) is an \(n \times n\) matrix. Show that if \(\mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \mathbf { x } _ { 3 }\) are linearly dependent then the vectors \(\mathbf { M x } _ { 1 } , \mathbf { M x } _ { 2 } , \mathbf { M x } _ { 3 }\) are also linearly dependent. The vectors \(\mathbf { y } _ { 1 } , \mathbf { y } _ { 2 } , \mathbf { y } _ { 3 }\) and the matrix \(\mathbf { P }\) are defined as follows: $$\begin{gathered} \mathbf { y } _ { 1 } = \left( \begin{array} { l } 1
5
7 \end{array} \right) , \quad \mathbf { y } _ { 2 } = \left( \begin{array} { r } 2
- 3

The matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that if \(\mathbf { A }\) is non-singular then

1. \(\lambda \neq 0\),
2. the matrix \(\mathbf { A } ^ { - 1 }\) has \(\lambda ^ { - 1 }\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & 2 \\ 0 & - 2 & 4 \\ 0 & 0 & - 3 \end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + 4 \mathbf { I } ) ^ { - 1 }$$ Find a non-singular matrix \(\mathbf { P }\), and a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { B } = \mathbf { P D P } ^ { - 1 }\).

4 The 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. Given that one eigenvalue of \(\mathbf { A }\) is 3 , find an eigenvalue of the matrix \(\mathbf { A } ^ { 4 } + 3 \mathbf { A } ^ { 2 } + 2 \mathbf { I }\), justifying your answer.

10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 }$$ where \(\lambda\) and \(\mu\) are real numbers and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\). 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 $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0
2 & 0 & 2
1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of \(\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. 8 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\). Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has equation \(2 x - y + z = 10\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). 9 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the coordinates of the centroid of the region bounded by \(C\), the \(x\)-axis and the line \(x = 1\). 10 The curve \(C\) has equation $$y = \frac { p x ^ { 2 } + 4 x + 1 } { x + 1 }$$ where \(p\) is a positive constant and \(p \neq 3\).

1. Obtain the equations of the asymptotes of \(C\).
2. Find the value of \(p\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
3. For the case \(p = 1\), show that \(C\) has no turning points, and sketch \(C\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.

The square matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that if \(\mathbf { A }\) is non-singular then

1. \(\lambda \neq 0\),
2. the matrix \(\mathbf { A } ^ { - 1 }\) has \(\lambda ^ { - 1 }\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. The \(3 \times 3\) matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 2 & - 4 \\ 0 & - 1 & 5 \\ 0 & 0 & 3 \end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + 3 \mathbf { I } ) ^ { - 1 }$$ where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. Find a non-singular matrix \(\mathbf { P }\), and a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { B } = \mathbf { P D P } ^ { - 1 }\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

The square matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that if \(\mathbf { A }\) is non-singular then

1. \(\lambda \neq 0\),
2. the matrix \(\mathbf { A } ^ { - 1 }\) has \(\lambda ^ { - 1 }\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. The \(3 \times 3\) matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 2 & - 4 \\ 0 & - 1 & 5 \\ 0 & 0 & 3 \end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + 3 \mathbf { I } ) ^ { - 1 }$$ where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. Find a non-singular matrix \(\mathbf { P }\), and a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { B } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

8 The vector \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A }\), with corresponding eigenvalue \(\lambda\), and is also an eigenvector of the matrix \(\mathbf { B }\), with corresponding eigenvalue \(\mu\). Show that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A B }\) with corresponding eigenvalue \(\lambda \mu\). State the eigenvalues of the matrix \(\mathbf { C }\), where $$\mathbf { C } = \left( \begin{array} { r r r } - 1 & - 1 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 2 \end{array} \right) ,$$ and find corresponding eigenvectors. Show that \(\left( \begin{array} { l } 1 \\ 6 \\ 3 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { D }\), where $$\mathbf { D } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ - 6 & - 3 & 4 \\ - 9 & - 3 & 7 \end{array} \right) ,$$ and state the corresponding eigenvalue. Hence state an eigenvector of the matrix CD and give the corresponding eigenvalue.

The vector \(\mathbf { e }\) is an eigenvector of each of the \(n \times n\) matrices \(\mathbf { A }\) and \(\mathbf { B }\), with corresponding eigenvalues \(\lambda\) and \(\mu\) respectively. Prove that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A B }\) with eigenvalue \(\lambda \mu\). It is given that the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 2 & 2 \\ - 2 & - 2 & - 2 \\ 1 & 2 & 2 \end{array} \right) ,$$ has eigenvectors \(\left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { r } 1 \\ 0 \\ - 1 \end{array} \right)\). Find the corresponding eigenvalues. Given that 2 is also an eigenvalue of \(\mathbf { A }\), find a corresponding eigenvector. The matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { r r r } - 1 & 2 & 2 \\ 2 & 2 & 2 \\ - 3 & - 6 & - 6 \end{array} \right) ,$$ has the same eigenvectors as \(\mathbf { A }\). Given that \(\mathbf { A B } = \mathbf { C }\), find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { - 1 } \mathbf { C } ^ { 2 } \mathbf { P } = \mathbf { D }$$

3 The transformation T is defined by the matrix \(\mathbf { M }\). The transformation S is defined by the matrix \(\mathbf { M } ^ { - 1 }\). Given that the point \(( x , y )\) is invariant under transformation T , prove that \(( x , y )\) is also an invariant point under transformation S .
[0pt] [3 marks]