Inverse Matrix Method (2x2)

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Marie Nicholson

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Marie Nicholson 7 years ago

Created this as a copy of Andrew's copy of Matrices: Inverse Matrix Method .

Name	Status	Author	Last Modified
Solve a pair of linear equations using a matrix	Ready to use	Bill Foster	01/06/2016 09:47
Unit 9: Question 8	Needs to be tested	rhaana starling	18/01/2022 17:04
Katie's copy of Solve a pair of linear equations using a matrix	Ready to use	Katie Dexter	01/06/2016 09:47
Andrew's copy of Matrices: Inverse Matrix Method	draft	Andrew Dunbar	30/06/2017 18:14
Inverse Matrix Method (2x2)	draft	Marie Nicholson	11/10/2021 11:22
Shaheen's copy of Andrew's copy of Matrices: Inverse Matrix Method	draft	Shaheen Charlwood	19/02/2020 23:02
2x2 Matrix Problems	draft	Tamsin Smith	17/09/2024 13:14

There are 22 other versions that do you not have access to.

Question statement

Write the following equations as a matrix equation

$Av=b$ for a matrix

$A$ and column vectors

$v$ and

$b$

$\begin{eqnarray*} \simplify[std]{{a}x+{b}y}&=&\var{c}\\ \simplify[std]{{a1}x+{b1}y}&=&\var{c1} \end{eqnarray*}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-7
addortakeaway	string	take away the equation
c	integer	8
b	integer	7
da	decimal	dec("28")
a1	decimal	dec("-7")
sc	integer	1
ab	number	7
sb1	integer	1
b1	integer	3
test	decimal	dec("-3")
sb	integer	1
sa	integer	-1
sc1	integer	-1
sa1	integer	-1
c1	integer	-4
inc	integer	-4

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
sa*random(1..9)

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`

-7

← Depends on:

→ Used by:

da
test

Statement
Advice
"Part a)" → "Gap 0." - Correct answer
"Part b)" → "Gap 0." - Correct answer
"Part d)" → "Gap 0." - Correct answer
"Part d)" → "Gap 1." - Correct answer

Parts

Part a) Gap-fill
1. Gap Gap 0. Matrix entry
  Add
2. Gap Gap 1. Mathematical expression
  Add
3. Gap Gap 2. Mathematical expression
  Add
4. Gap Gap 3. Matrix entry
  Add
Part b) Gap-fill
1. Gap Gap 0. Matrix entry
  Add
2. Gap Gap 1. Mathematical expression
  Add
Part c) Gap-fill
1. Gap Matrix entry
  Add
Part d) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add

Gap-fill

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

$A =$ Gap 0.

$v = \;\;\Bigg($	Gap 1.	$\Bigg)$
	Gap 2.

$b =$ Gap 3.

Advice

a)

The equations can be written in the matrix form:

$\begin{pmatrix} \var{a} & \var{b}\\ \var{a1}&\var{b1} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \var{c} \\ \var{c1} \end{pmatrix}$

b)

Since $\mathrm{det}(A) = \simplify[]{{a}*{b1}-{b}*{a1}={dA}} \neq 0$ , $A$ is invertible and

$A^{-1} = \begin{pmatrix} \simplify[std]{{b1}/{dA}}&\simplify[std]{{-b}/{dA}}\\\simplify[std]{{-a1}/{dA}}&\simplify[std]{{a}/{dA}} \end{pmatrix}$

c)

We have:

$\begin{eqnarray*} A^{-1}b &=& \begin{pmatrix} \simplify[std]{{b1}/{dA}}&\simplify[std]{{-b}/{dA}}\\\simplify[std]{{-a1}/{dA}}&\simplify[std]{{a}/{dA}} \end{pmatrix}\begin{pmatrix} \var{c}\\\var{c1}\end{pmatrix} \\ &=& \begin{pmatrix} \simplify[std]{{c*b1-c1*b}/{dA}}\\\simplify[std]{{c1*a-c*a1}/{dA}}\end{pmatrix} \end{eqnarray*}$

d)

Note that $Av = b \Rightarrow v = A^{-1}b$ hence we can read the solution from the last part as this gives:

$\begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} \simplify[std]{{c*b1-c1*b}/{dA}}\\ \simplify[std]{{c1*a-c*a1}/{dA}}\end{pmatrix}$

Hence

$\begin{eqnarray*} x&=& \simplify[std]{{c*b1-c1*b}/{dA}}\\ y&=& \simplify[std]{{c1*a-c*a1}/{dA}} \end{eqnarray*}$

Give a worked solution to the whole question.

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This question is used in the following exam:

Solving systems of equations by Marie Nicholson in Marie's linear algebra workspace.

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Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

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Contributors

History

Comment

Checkpoint description

Marie Nicholson 7 years ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

b)

c)

d)

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Inverse Matrix Method (2x2)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Marie Nicholson 7 years ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

b)

c)

d)

Generated value: `integer`