Invariant lines and eigenvalues and vectors

189 questions · 24 question types identified

Sort by: Question count | Difficulty
Find P and D for A = PDP⁻¹

Questions asking to find matrices P and diagonal matrix D such that A = PDP⁻¹ (standard diagonalization of matrix A)

26 Standard +0.6
13.8% of questions
Show example »
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 \(\mathbf { D } = \mathbf { P } ^ { - 1 } \mathbf { A P }\)
View full question →
Easiest question Standard +0.3 »
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)
View full question →
Hardest question Challenging +1.8 »
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.
View full question →
Find line of invariant points

A question is this type if and only if it asks to find the equation of a line where every point on the line is mapped to itself (invariant points, not just invariant direction).

20 Standard +0.5
10.6% of questions
Show example »
1 A surface is defined in 3-D by \(z = 3 x ^ { 3 } + 6 x y + y ^ { 2 }\).
Determine the coordinates of any stationary points on the surface.
View full question →
Easiest question Moderate -0.3 »
3 Find the equation of the line of invariant points under the transformation given by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 3 & - 1 \\ 2 & 0 \end{array} \right)\).
View full question →
Hardest question Challenging +1.8 »
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 .
View full question →
Prove eigenvalue/eigenvector properties

A question is this type if and only if it asks to prove general properties about eigenvalues and eigenvectors (e.g., if λ is eigenvalue of A then λ² is eigenvalue of A²).

17 Standard +0.7
9.0% of questions
Show example »
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]
View full question →
Easiest question Standard +0.3 »
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.
View full question →
Hardest question Challenging +1.2 »
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.
View full question →
Find eigenvectors given eigenvalue

A question is this type if and only if it provides an eigenvalue and asks to find a corresponding eigenvector, or verifies that a given vector is an eigenvector.

16 Standard +0.1
8.5% of questions
Show example »
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.
View full question →
Easiest question Moderate -0.8 »
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.
View full question →
Hardest question Challenging +1.2 »
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.
    OCR 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. }
View full question →
Find invariant lines through origin

A question is this type if and only if it asks to find the equations of invariant lines (not necessarily lines of invariant points) through the origin for a 2×2 transformation matrix.

16 Standard +0.7
8.5% of questions
Show example »
6 Find the invariant line of the transformation of the \(x - y\) plane represented by the matrix \(\left( \begin{array} { r r } 2 & 0 \\ 4 & - 1 \end{array} \right)\).
View full question →
Easiest question Moderate -0.5 »
6 Find the invariant line of the transformation of the \(x - y\) plane represented by the matrix \(\left( \begin{array} { r r } 2 & 0 \\ 4 & - 1 \end{array} \right)\).
View full question →
Hardest question Challenging +1.2 »
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 \(\mathbf { C A B }\).
View full question →
Find eigenvalues/vectors of matrix combination

A question is this type if and only if it asks to find eigenvalues or eigenvectors of a matrix expression like A+kI, A⁻¹, AB, or A²+bA+cI given eigenvalues of A.

12 Standard +0.4
6.3% of questions
Show example »
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 \(\mathbf { A } + \mathbf { A } ^ { 2 }\).
View full question →
Easiest question Standard +0.3 »
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 \(\mathbf { A } + \mathbf { A } ^ { 2 }\).
View full question →
Hardest question Standard +0.8 »
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 \(\mathbf { B } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
View full question →
Find P and D for A² = PDP⁻¹ or A⁻¹ = PDP⁻¹

Questions asking to find matrices P and diagonal matrix D such that A² = PDP⁻¹ or A⁻¹ = PDP⁻¹ (diagonalization of a power or inverse of A)

11 Challenging +1.3
5.8% of questions
Show example »
7 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } - 6 & 2 & 13 \\ 0 & - 2 & 5 \\ 0 & 0 & 8 \end{array} \right) .$$
  1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { - 1 } = \mathbf { P D P } ^ { - 1 }\).
  2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).
View full question →
Easiest question Standard +0.8 »
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 \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
View full question →
Hardest question Challenging +1.8 »
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.
View full question →
Orthogonal matrix diagonalization

A question is this type if and only if it asks to find an orthogonal matrix P and diagonal matrix D such that P^T·M·P = D, typically for symmetric matrices.

6 Standard +0.7
3.2% of questions
Show example »
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 \(\mathbf { P } ^ { \mathrm { T } } \mathbf { A P } = \mathbf { D }\).
View full question →
Use Cayley-Hamilton for matrix power

A question is this type if and only if it asks to use the Cayley-Hamilton theorem to express a high power of a matrix (e.g., A⁴) in terms of lower powers.

6 Standard +0.6
3.2% of questions
Show example »
  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 \(\mathbf { A } ^ { 3 } = 43 \mathbf { A } - 42 \mathbf { I }\).
View full question →
Use Cayley-Hamilton for inverse

A question is this type if and only if it asks to use the Cayley-Hamilton theorem to find the inverse of a matrix expressed in terms of powers of the matrix.

6 Standard +0.3
3.2% of questions
Show example »
3 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { l l l } 6 & - 9 & 5 \\ 5 & - 8 & 5 \\ 1 & - 1 & 2 \end{array} \right)$$
  1. Find the eigenvalues of \(\mathbf { A }\).
  2. Use the characteristic equation of \(\mathbf { A }\) to show that \(\mathbf { A } ^ { - 1 } = p \mathbf { A } ^ { 2 } + q \mathbf { l }\), where \(p\) and \(q\) are constants to be determined.
View full question →
Matrix powers by induction

A question is this type if and only if it asks to prove a formula for M^n by mathematical induction.

5 Standard +0.3
2.6% of questions
Show example »
6 The matrix \(\mathbf { M }\) is \(\left( \begin{array} { r r } 2 & 1 \\ - 1 & 0 \end{array} \right)\).
  1. Calculate \(\mathbf { M } ^ { 2 } , \mathbf { M } ^ { 3 }\) and \(\mathbf { M } ^ { 4 }\).
  2. Hence make a conjecture about the matrix \(\mathbf { M } ^ { n }\).
  3. Prove your conjecture.
View full question →
Find constant from eigenvalue condition

Questions asking to find unknown constants in a matrix given that a specific eigenvalue exists or that a given vector is an eigenvector

4 Standard +0.4
2.1% of questions
Show example »
  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 \(T\).
View full question →
Find eigenvalues of 3×3 matrix

A question is this type if and only if it asks to find or verify the eigenvalues of a 3×3 matrix, possibly including showing the characteristic equation.

4 Standard +0.4
2.1% of questions
Show example »
3 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 1 & 7 \\ 0 & 6 & 0 \\ 7 & 7 & 5 \end{array} \right) .$$
  1. Find the eigenvalues of \(\mathbf { A }\).
  2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-06_568_1614_294_262} The diagram shows the curve with equation \(\mathrm { y } = \ln \mathrm { x }\) for \(x \geqslant 1\), together with a set of ( \(N - 1\) ) rectangles of unit width.
  3. By considering the sum of the areas of these rectangles, show that $$\ln N ! > N \ln N - N + 1 .$$
  4. Use a similar method to find, in terms of \(N\), an upper bound for \(\operatorname { In } N\) !.
View full question →
Find eigenvalues of 2×2 matrix

A question is this type if and only if it asks to find the eigenvalues of a 2×2 matrix without requiring eigenvectors or further diagonalization.

4 Standard +0.4
2.1% of questions
Show example »
3 The surface with equation \(z = x ^ { 3 } + y ^ { 3 } - 6 x y\) has two stationary points; one at the origin and the second at the point \(A\). Determine the coordinates of \(A\).
View full question →
Find constant from invariant line or area condition

Questions asking to find unknown constants in a matrix given conditions about invariant lines, area scaling, or other geometric transformation properties

3 Standard +0.6
1.6% of questions
Show example »
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.
View full question →
Find matrix A given eigenvalues and eigenvectors

Questions asking to construct/find a matrix A given its eigenvalues and corresponding eigenvectors

3 Standard +0.8
1.6% of questions
Show example »
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.
View full question →
Normalized eigenvectors

A question is this type if and only if it specifically asks to find normalized (unit length) eigenvectors.

3 Standard +0.5
1.6% of questions
Show example »
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
View full question →
Verify invariant line property

A question is this type if and only if it asks to show or verify that a given line (e.g., y = kx) is invariant under a transformation, or to find k such that a line is invariant.

3 Standard +0.3
1.6% of questions
Show example »
6 Given that \(y = m x\) is an invariant line of the transformation with matrix \(\left( \begin{array} { r r } 1 & 2 \\ 2 & - 2 \end{array} \right)\), determine the possible values of \(m\). Section B (113 marks)
Answer all the questions.
View full question →
Describe geometric transformation from matrix

A question is this type if and only if it asks to identify or describe the geometric transformation(s) represented by a given matrix (rotation, reflection, stretch, shear, etc.).

2 Challenging +1.5
1.1% of questions
Show example »
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.
View full question →
Find constant from singularity condition

Questions asking to find unknown constants in a matrix given that the matrix is singular (determinant = 0) or a system has no unique solution

0
0.0% of questions
Find inverse transformation matrix

A question is this type if and only if it asks to find the matrix that reverses a given transformation (maps image back to original).

0
0.0% of questions
Composite transformation matrix

A question is this type if and only if it asks to find the single matrix representing a sequence of transformations by multiplying transformation matrices.

0
0.0% of questions
Find matrix from geometric description

A question is this type if and only if it asks to find the matrix representing a described sequence of geometric transformations.

0
0.0% of questions
Area scaling under transformation

A question is this type if and only if it asks to find the area of an image shape given the area of the original shape and a transformation matrix, using determinants.

0
0.0% of questions
Unclassified

Questions not yet assigned to a type.

22
11.6% of questions
Show 22 unclassified »
6 Let \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0 \\ 1 & 1 \end{array} \right)\).
  1. The transformation in the \(x - y\) plane represented by \(\mathbf { A } ^ { - 1 }\) transforms a triangle of area \(30 \mathrm {~cm} ^ { 2 }\) into a triangle of area \(d \mathrm {~cm} ^ { 2 }\). Find the value of \(d\).
  2. Prove by mathematical induction that, for all positive integers \(n\), $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 0 \\ 2 ^ { n } - 1 & 1 \end{array} \right)$$
  3. The line \(y = 2 x\) is invariant under the transformation in the \(x - y\) plane represented by \(\mathbf { A } ^ { n } \mathbf { B }\), where \(\mathbf { B } = \left( \begin{array} { r l } 1 & 0 \\ 33 & 0 \end{array} \right)\). Find the value of \(n\).
4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } \mathrm { a } & \mathrm { b } ^ { 2 } \\ \mathrm { c } ^ { 2 } & \mathrm { a } \end{array} \right)\), where \(a , b , c\) are real constants and \(b \neq 0\).
  1. Show that \(\mathbf { M }\) does not represent a rotation about the origin.
  2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { M }\).
    It is given that \(\mathbf { M }\) represents the sequence of two transformations in the \(x - y\) plane given by an enlargement, centre the origin, scale factor 5 followed by a shear, \(x\)-axis fixed, with \(( 0,1 )\) mapped to \(( 5,1 )\).
  3. Find \(\mathbf { M }\).
  4. The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(P Q R\). Given that the area of triangle \(D E F\) is \(12 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(P Q R\).
4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \sqrt { 3 } \\ \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } \end{array} \right) \left( \begin{array} { c c } 14 & 0 \\ 0 & 1 \end{array} \right)\).
  1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations in the \(x - y\) plane. Give full details of each transformation, and make clear the order in which they are applied.
  2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
  3. Find the equations of the invariant lines, through the origin, of the transformation represented by \(\mathbf { M }\).
  4. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Given that the area of triangle \(D E F\) is \(28 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(A B C\).
1 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 1 & b \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } a & 0 \\ 0 & 1 \end{array} \right)\), where \(a\) and \(b\) are positive constants.
  1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
    The unit square in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto parallelogram \(O P Q R\).
  2. Find, in terms of \(a\) and \(b\), the matrix which transforms parallelogram \(O P Q R\) onto the unit square.
    It is given that the area of \(O P Q R\) is \(2 \mathrm {~cm} ^ { 2 }\) and that the line \(\mathrm { x } + 3 \mathrm { y } = 0\) is invariant under the transformation represented by \(\mathbf { M }\).
  3. Find the values of \(a\) and \(b\).
5 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & - \frac { 1 } { 2 } \sqrt { 2 } \\ \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right) \left( \begin{array} { c c } 1 & k \\ 0 & 1 \end{array} \right)\), where \(k\) is a constant.
  1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
  2. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Find, in terms of \(k\), the single matrix which transforms triangle \(D E F\) onto triangle \(A B C\).
  3. Find the set of values of \(k\) for which the transformation represented by \(\mathbf { M }\) has no invariant lines through the origin.
5 Let \(k\) be a constant. The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & k & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } 0 & - 2 \\ - 1 & 3 \\ 0 & 0 \end{array} \right) \quad \text { and } \quad \mathbf { C } = \left( \begin{array} { r r r } - 2 & - 1 & 1 \\ 1 & 1 & 3 \end{array} \right)$$ It is given that \(\mathbf { A }\) is singular.
  1. Show that \(\mathbf { C A B } = \left( \begin{array} { r r } 3 & - 7 \\ - 9 & 3 \end{array} \right)\).
  2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\).
  3. The matrices \(\mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\) represent geometrical transformations in the \(x - y\) plane.
    • D represents an enlargement, centre the origin.
    • E represents a stretch parallel to the \(x\)-axis.
    • F represents a reflection in the line \(y = x\).
    Given that \(\mathbf { C A B } = \mathbf { D } - 9 \mathbf { E F }\), find \(\mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\).
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 \(\mathbf { A } ^ { 5 } = \mathbf { P D Q }\).
8 Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A } = \left( \begin{array} { r r r } 4 & - 1 & 1 \\ - 1 & 0 & - 3 \\ 1 & - 3 & 0 \end{array} \right)\). Find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
6 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { l l l } 4 & - 5 & 3 \\ 3 & - 4 & 3 \\ 1 & - 1 & 2 \end{array} \right)$$ Show that \(\mathbf { e } = \left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\) and state the corresponding eigenvalue. Find the other two eigenvalues of \(\mathbf { A }\). The matrix \(\mathbf { B }\) is given by $$\mathbf { B } = \left( \begin{array} { r r r } - 1 & 4 & 0 \\ - 1 & 3 & 1 \\ 1 & - 1 & 3 \end{array} \right)$$ Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { B }\) and deduce an eigenvector of the matrix \(\mathbf { A B }\), stating the corresponding eigenvalue.
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.]
5 The matrix \(\mathbf { A }\), given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & - 2 \\ 6 & 4 & - 6 \\ 6 & 5 & - 7 \end{array} \right)$$ has eigenvalues \(1 , - 1\) and - 2 .
  1. Find a set of corresponding eigenvectors.
  2. The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \mathbf { A } - 2 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. Write down the eigenvalues of \(\mathbf { B }\), and state a set of corresponding eigenvectors.
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 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$y = \frac { a x } { x + 5 } \quad \text { and } \quad y = \frac { x ^ { 2 } + ( a + 10 ) x + 5 a + 26 } { x + 5 }$$ respectively, where \(a\) is a constant and \(a > 2\).
  1. Find the equations of the asymptotes of \(C _ { 1 }\).
  2. Find the equation of the oblique asymptote of \(C _ { 2 }\).
  3. Show that \(C _ { 1 }\) and \(C _ { 2 }\) do not intersect.
  4. Find the coordinates of the stationary points of \(C _ { 2 }\).
  5. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on a single diagram. [You do not need to calculate the coordinates of any points where \(C _ { 2 }\) crosses the axes.]
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 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. }
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 \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\).
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.
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.
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.
  1. 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.
  2. 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.
  3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 3 } = \mathbf { P D P } ^ { - 1 }\).
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 \(\mathbf { M } ^ { 7 } \mathbf { P }\).
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 \(\mathbf { M } ^ { 7 } \mathbf { P }\).
6 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } 2 & - 3 & - 7 \\ 0 & 5 & 7 \\ 0 & 0 & - 2 \end{array} \right) .$$
  1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
  2. Use the characteristic equation of \(\mathbf { A }\) to show that $$\mathbf { A } ^ { 4 } = a \mathbf { A } ^ { 2 } + b \mathbf { I } ,$$ where \(a\) and \(b\) are integers to be determined.