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Tamsin Smith 6 months, 1 week ago

Published this.

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Tamsin Smith 8 months, 3 weeks ago

Created this as a copy of Inverse Matrix Method (2x2).

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

$2\times2$ Matrix Problems

Solve the set of simultaneous equations using matrices:

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-5
c	integer	-5
b	integer	-5
a1	integer	4
b1	integer	-5
c1	integer	-2
det	integer	45

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(-5..5 except 0)

JME function reference

Description

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

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Generated value: `integer`

-5

→ Used by:

This variable doesn't seem to be used anywhere.

Parts

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

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

Write the following set of equations as a matrix equation:

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

That is, in the form:

$\boldsymbol{A}v=b$

for some matrix $\boldsymbol{A}$ and column vectors $v$ and $b$ .

Gap 0. $\begin{pmatrix} x \\ y \\ \end{pmatrix} = \$ Gap 1.

Advice

By using the coefficients of the variables in the equations, we can write the equations as:

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

Notice that carrying out the matrix multiplication will give you your system of equations.

We first find the determinant of $\boldsymbol{A}$ :

$\det \boldsymbol{A} = \var{a} \times \var{b1} - \var{b} \times \var{a1} = \var{det}$

Since $\det \boldsymbol{A} \neq 0$ , $A$ is invertible with:

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

We have the equation:

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

Using standard matrix multiplication, we can compute:

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

Notice that

$\boldsymbol{A}v = b \Rightarrow v = \boldsymbol{A}^{-1}b$

Hence, we can use our solution to part $c)$ to find:

$\begin{pmatrix} x \\ y \end{pmatrix} = \boldsymbol{A}^{-1}b = \begin{pmatrix} \simplify[std]{{c*b1-c1*b}/{det}} \\ \simplify[std]{{c1*a-c*a1}/{det}} \end{pmatrix}$

And therefore, we have:

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

Give a worked solution to the whole question.

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2x2 Matrix Problems

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Tamsin Smith 6 months, 1 week ago

Tamsin Smith 8 months, 3 weeks ago

$2\times2$ Matrix Problems

Error in variable testing condition

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Generated value: `integer`

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Gap-fill

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2x2 Matrix Problems

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Tamsin Smith 6 months, 1 week ago

Tamsin Smith 8 months, 3 weeks ago

2×22\times2 Matrix Problems

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

$2\times2$ Matrix Problems

Generated value: `integer`