Solve simultaneous equations by finding inverse matrix

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Morteza Shahpari 6 years, 5 months ago

Created this as a copy of Blathnaid's copy of Solve simultaneous equations by finding inverse matrix,.

Name	Status	Author	Last Modified
Solve simultaneous equations by finding inverse matrix,	Ready to use	Newcastle University Mathematics and Statistics	21/12/2022 13:36
Blathnaid's copy of Solve simultaneous equations by finding inverse matrix,	draft	Blathnaid Sheridan	12/11/2018 13:23
Solve simultaneous equations by finding inverse matrix	draft	Morteza Shahpari	20/11/2018 05:28
Morteza's copy of Solve simultaneous equations by Crammer's method	draft	Morteza Shahpari	23/11/2018 06:03
Simon's copy of Solve simultaneous equations by finding inverse matrix,	draft	Simon Thomas	01/04/2019 15:36
Inverse of 2x2 matrix	draft	Peter Johnston	16/04/2019 03:48
sean's copy of Simon's copy of Solve simultaneous equations by finding inverse matrix,	draft	sean hunte	07/05/2019 20:25
Maria's copy of Solve simultaneous equations by finding inverse matrix,	draft	Maria Aneiros	23/05/2019 07:00
W1b - Solve simultaneous equations by finding inverse matrix	Ready to use	Timur Zaripov	17/06/2022 09:37
Alexander's copy of Solve simultaneous equations by finding inverse matrix,	draft	Alexander Corner	22/10/2024 10:09
M2	Ready to use	Shaheen Charlwood	16/05/2022 14:24

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Question statement

Rewrite the following system of equations as a matrix equation

$\mathbf{Av} = \mathbf{b}$

for a matrix $\mathbf{A}$ and column vectors $\mathbf{v}$ and $\mathbf{b}$ .

$\begin{align} \simplify[std]{ {ma[0][0]}x + {ma[0][1]}y} &= \var{mb[0][0]} \\ \simplify[std]{ {ma[1][0]}x + {ma[1][1]}y} &= \var{mb[1][0]} \end{align}$

Input all numbers as fractions or integers and not as decimals.

Give any introductory information the student needs.

			Name	Type	Generated Value

ma	matrix	matrix([2,2],[-4,-1])
a00	integer	2
a01	integer	2
a10	integer	-4
a11	integer	-1
mb	matrix	matrix([3],[-7])
ma_inverse	matrix	Matrix of size 2×2
x	number	1.8333333333
y	number	-0.3333333333

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
matrix([
    [a00,a01],
    [a10,a11]
])

JME function reference

Description

Matrix A. a10 is picked so it's non-singular, and a11 is never $\pm a01$ .

No entry is 0.

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

Can an exam override the value of this variable?

Generated value: `matrix`

matrix([2,2],[-4,-1])

← Depends on:

This variable doesn't seem to be used anywhere.

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 Matrix entry
  Add
Part c) Gap-fill
1. Gap Matrix entry
  Add
Part d) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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Gap-fill

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

$\mathbf{A} =$ Gap 0.

$\mathbf{v} =$

Gap 1.

Gap 2.

$\mathbf{b} =$ Gap 3.

Advice

a)

The equations can be written in the matrix form

$\var{ma}\begin{pmatrix} x \\ y \end{pmatrix} = \var{mb}$

b)

$\mathrm{det}(\mathbf{A}) = \simplify[]{ {ma[0][0]}*{ma[1][1]} - {ma[0][1]}*{ma[1][0]}} = \var{det(ma)} \neq 0$ , so $\mathbf{A}$ is invertible.

$\mathbf{A}^{-1} = \simplify[fractionnumbers]{{ma_inverse}}$

c)

We have

$\begin{align} \mathbf{A}^{-1}\mathbf{b} &= \simplify[fractionnumbers]{{ma_inverse}*{mb}} \\ &= \simplify[fractionnumbers]{{ma_inverse*mb}} \end{align}$

d)

Rearrange the equation $\mathbf{Av}=\mathbf{b}$ to make $\mathbf{v}$ the subject:

$\begin{align} \mathbf{A}^{-1}\mathbf{A}\mathbf{v} &= \mathbf{A}^{-1}\mathbf{b} \\ \mathbf{v} &= \mathbf{A}^{-1}\mathbf{b} \\ \\ \end{align}$

Hence,

$\begin{pmatrix} x \\ y \end{pmatrix} = \simplify[fractionnumbers]{{ma_inverse*mb}}$

That is,

$\begin{align} x &= \simplify[fractionnumbers]{{x}}, \\ \\ y &= \simplify[fractionnumbers]{{y}} \end{align}$

Give a worked solution to the whole question.

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This question is used in the following exam:

Linear algebra for electric circuits 1301ENG by Morteza Shahpari in Linear algebra for electric circuits 1301ENG.

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Solve simultaneous equations by finding inverse matrix

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Morteza Shahpari 6 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: `matrix`

Parts

Gap-fill

a)

b)

c)

d)

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Error

Solve simultaneous equations by finding inverse matrix

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Morteza Shahpari 6 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: matrix

Parts

Gap-fill

a)

b)

c)

d)

Generated value: `matrix`