Ch10 Q5

Error

There was an error loading the page.

Contributors

Alex Van den Hof

History

○

Alex Van den Hof 8 years, 2 months ago

Created this as a copy of Alex's copy of Simon's copy of Find eigenvalues and eigenvectors of matrices, and power of one matrix.

Name	Status	Author	Last Modified
Find eigenvalues and eigenvectors of matrices, and power of one matrix	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
First Order Differential Equations - Coupled FINISHED	draft	Gareth Woods	26/08/2016 14:36
Gareth's copy of Find eigenvalues and eigenvectors of matrices, and power of one matrix	draft	Gareth Woods	26/08/2016 11:21
Simon's copy of Find eigenvalues and eigenvectors of matrices, and power of one matrix	draft	Simon Vaughan	09/11/2016 16:37
Alex's copy of Simon's copy of Find eigenvalues and eigenvectors of matrices, and power of one matrix	draft	Alex Van den Hof	20/02/2017 15:52
Ch10 Q5	draft	Alex Van den Hof	20/02/2017 16:05
Harry's copy of Find eigenvalues and eigenvectors of matrices, and power of one matrix	draft	Harry Flynn	09/03/2018 14:47
Simon's copy of Find eigenvalues and eigenvectors of matrices, and power of one matrix	draft	Simon Thomas	12/06/2019 09:53
Priyanka's copy of Find eigenvalues and eigenvectors of matrices, and power of one matrix	draft	Priyanka Pandya	10/02/2020 16:21
Eigenvalues	draft	Heather Driscoll	06/05/2021 10:51
Eigenvectors	draft	Heather Driscoll	21/04/2021 15:31
Ugur's copy of Find eigenvalues and eigenvectors of matrices, and power of one matrix	Ready to use	Ugur Efem	13/10/2024 20:16
Ugur's copy of Eigenvalues	Ready to use	Ugur Efem	13/10/2024 20:15

There are 25 other versions that do you not have access to.

Question statement

Find the eigenvalues and eigenvectors for the matrix $A$ where:

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

Give any introductory information the student needs.

			Name	Type	Generated Value

test1	integer	-5
a21	number	22
a22	integer	-3
bn22	integer	486
bn21	integer	405
b22	integer	-4
b21	integer	-1
b1	integer	-3
cn11	number	2673
mna	integer	-5
mnb	integer	-6
cn12	number	2430
s2	integer	1
s1	integer	-1
a12	integer	1
b12	number	-2
b11	integer	-5
test	integer	6
cn21	number	1215
cn22	number	1458
a11	integer	6
that	integer	2
db	number	18
bn12	integer	810
bn11	integer	891
da	number	-40
a1	integer	-6
x2	integer	-1
c2	integer	-6
c1	integer	8
x1	integer	2
mxb	integer	-3
mxa	integer	8
a	integer	-5
b	integer	8
f	number	3
tra	integer	3
trb	integer	-9
n	integer	4
this	integer	4
s	integer	1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(-8..8)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

-5

→ Used by:

This variable doesn't seem to be used anywhere.

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 Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add

Gap-fill

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

Find the eigenvalues of $A$ .

Let $a_1$ be the least eigenvalue of $A,\;\;\; a_1=\;\;$ Gap 0.

Let $a_2$ be the greatest eigenvalue of $A,\;\; a_2=\;\;$ Gap 1.

Advice

Matrix $A$

$A - \lambda I_2 = \begin{pmatrix} \var{a11}-\lambda & \var{a12}\\ \var{a21} & \var{a22}-\lambda \end{pmatrix}$
Hence the characteristic polynomial

$p(\lambda)$ is:

$\begin{eqnarray*} \mathrm{det}\left(A-\lambda I_2 \right)&=&\simplify[zeroTerm]{({a11}-lambda)({a22}-lambda)-{a12}*{a21}}\\ &=& \simplify[std]{lambda^2-{trA}*lambda+{dA}}\\ &=&\simplify[std]{(lambda-{a})(lambda-{b})} \end{eqnarray*}$
We see that on solving

$p(\lambda)=0$ we get the eigenvalues:

$\lambda_1=\var{mnA},\;\;\;\lambda_2=\var{mxA}$
Note: We could have found the characteristic polynomial by noting that for a 2 × 2 matrix

$A$ then the characteristic polynomial is

$\lambda^2-\mathrm{trace}(A)+\mathrm{det}(A)$
where

$\mathrm{trace}(A) = \var{trA},\;\;\;\mathrm{det}(A)=\var{dA}$

Finding the eigenvectors:

1. $\lambda=\var{mnA}$

We have the eigenspace is given by all $v=(x,y)$ such that $(\simplify{A-{mnA}}I_2)v=(0,0)$ i.e.

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

This gives the two equations:

$\begin{eqnarray*} \simplify[std]{{a11-mnA}x + {a12}y}&=&0\\ \simplify[std]{{a21}x + {a22-mnA}y}&=&0 \end{eqnarray*}$
There is only one equation here as we see that the equations are the same (one is a multiple of the other).

So putting $x=1$ in the first equation we get $y_1=\var{-s*(a11-mnA)}$

Hence the eigenvector we want is

$\begin{pmatrix} 1 \\ \var{-s*(a11-mnA)} \end{pmatrix}$

2. $\lambda=\var{mxA}$

In this case we have the equations:

$\begin{eqnarray*} \simplify[std]{{a11-mxA}x + {a12}y}&=&0\\ \simplify[std]{{a21}x + {a22-mxA}y}&=&0 \end{eqnarray*}$

Once again there is only one equation, so putting $x=1$ in the first equation we get $y_2=\var{-s*(a11-mxA)}$

Hence the eigenvector we want is

$\begin{pmatrix} 1 \\ \var{-s*(a11-mxA)} \end{pmatrix}$

Give a worked solution to the whole question.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Number entry
Number entry		Hasn't run yet
Number entry
Number entry		Hasn't run yet

This question is used in the following exams:

Ch10 Q5 by Alex Van den Hof in LeicesterPhysPractice.
matrices Q5 by Simon Vaughan in LeicesterPhysPractice.
matrices Q5 by Simon Vaughan in Simon's workspace.

Loading...

Error

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Alex Van den Hof 8 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Matrix $A$

Finding the eigenvectors:

Loading...

Error

Ch10 Q5

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Alex Van den Hof 8 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Matrix AA

Finding the eigenvectors:

Generated value: `integer`

Matrix $A$