Maria's copy of Eigenvalues and one eigenvector of a 2x2 matrix

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Maria Aneiros 5 years, 10 months ago

Created this as a copy of Eigenvalues and one eigenvector of a 2x2 matrix.

Name	Status	Author	Last Modified
Eigenvalues & eigenvectors of a 2x2	Ready to use	Frank Doheny	14/03/2018 21:04
Owen's copy of Eigenvalues & eigenvectors of a 2x2	draft	Owen Jepps	15/05/2017 09:17
Elias Jakobus's copy of Owen's copy of Eigenvalues & eigenvectors of a 2x2	draft	Elias Jakobus Willemse	23/05/2017 12:58
Eigenvalues & eigenvectors	draft	Matthew Mears	27/03/2020 16:32
Eigenvalues and one eigenvector of a 2x2 matrix	draft	Peter Johnston	16/04/2020 21:49
Timothy's copy of Eigenvalues and one eigenvector of a 2x2 matrix	draft	Timothy Gould	16/04/2019 06:05
Maria's copy of Eigenvalues and one eigenvector of a 2x2 matrix	draft	Maria Aneiros	23/05/2019 06:32
Eigenvalues/Eigenvector of a 2x2 matrix	draft	Thomas Waters	12/09/2019 20:04

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Question statement

Given the matrix

$A =\begin{pmatrix} \var{a11}&\var{a12}\\ \var{a21}&\var{a22}\\ \end{pmatrix}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a11	integer	10
c1	integer	19
a12	integer	-54
k	integer	6
a21	integer	3
a22	integer	37
lambda1	integer	19
lambda2	integer	28

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A random number between and (inclusive) with step size

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Generated value: `integer`

→ Used by:

a12
lambda1
lambda2

Statement
Advice

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
Part b) Gap-fill
1. Gap Number entry
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Calculate the eigenvalues of the matrix A

$\lambda_1$ is the lesser of the two eigenvalues and $\lambda_2$ is the greater of the two eigenvalues;

$\lambda_1$ = Gap 0.

$\lambda_2$ = Gap 1.

Advice

The eigenvalues of a matrix are the values of $\lambda$ that satisfy the relation

$|A-\lambda I| = 0$

$\begin{vmatrix} \var{a11}-\lambda&\var{a12}\\ \var{a21}&\var{a22}-\lambda\\ \end{vmatrix}=0$

This gives:

$(\var{a11}-\lambda)*(\var{a22}-\lambda)-(\var{a12})*(\var{a21})=0$

$\lambda^2-\simplify{{a11}+{a22}}\lambda+\simplify{{a11}*{a22}-{a21}*{a12}}=0$

This can be solved using factorisation or by the quadratic formula to give:

$\lambda_1 =\var{lambda1}$ and $\lambda_2 =\var{lambda2}$

An eigenvector $v=\begin{pmatrix} x\\ y\\ \end{pmatrix}$ corresponding to an eigenvalue $\lambda$ must satisfy the relation: $(A-\lambda I)v = 0$

so for $\lambda_1=\var{lambda1}$

$\begin{pmatrix} \simplify{{a11}-{lambda1}}&\var{a12}\\ \var{a21}&\simplify{{a22}-{lambda1}}\\ \end{pmatrix}\begin{pmatrix} x\\ \var{a21}\\ \end{pmatrix}=0$

thus

$\var{a21}x+\simplify{{a22}-{lambda1}}*\var{a21}=0$

$\Rightarrow ~ \var{a21}x=-\simplify{({a22}-{lambda1})*{a21}}$

$\Rightarrow ~ x=-\simplify{({a22}-{lambda1})}$

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This question is used in the following exam:

Maths 1A - Homework Week 4 (Matrices) by Maria Aneiros in MY QUESTIONS.

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Maria's copy of Eigenvalues and one eigenvector of a 2x2 matrix

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Taxonomy: mathcentre

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Maria Aneiros 5 years, 10 months ago

Error in variable testing condition

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Warning

Generated value: `integer`

Parts

Gap-fill

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Error

Maria's copy of Eigenvalues and one eigenvector of a 2x2 matrix

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Maria Aneiros 5 years, 10 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`