Determinant and inverse of 2x2 matrices

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Michael Foreman

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Christian Lawson-Perfect

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8 years, 9 months ago

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Christian Lawson-Perfect 8 years, 9 months ago

Gave some feedback: Ready to use

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Michael Foreman 9 years, 2 months ago

Created this as a copy of Christoph's copy of Determinant and inverse of 2x2 matrices.

Name	Status	Author	Last Modified
Determinant and inverse of 2x2 matrices	Ready to use	Christian Lawson-Perfect	01/06/2016 11:03
Christoph's copy of Determinant and inverse of 2x2 matrices	Ready to use	Christoph Walti	01/06/2016 09:47
Determinant and inverse of 2x2 matrices	Ready to use	Michael Foreman	01/06/2016 09:47
Hina's copy of Determinant and inverse of 2x2 matrices	draft	Hina Ahmed	09/11/2016 13:32
Simon's copy of Hina's copy of Determinant and inverse of 2x2 matrices	draft	Simon Vaughan	09/11/2016 16:37
Alex's copy of Simon's copy of Hina's copy of Determinant and inverse of 2x2 matrices	draft	Alex Van den Hof	26/01/2017 14:17
Ch10 Q3	draft	Alex Van den Hof	20/02/2017 16:01
Ex 5 Determinant and Inverse of 2x2 matrices	Ready to use	Violeta CIT	21/09/2017 22:04
MATH6058 Determinant and Inverse of 2x2 matrices	draft	Catherine Palmer	25/09/2017 11:50
Question 4 MATH 6005 Assessment 1 Determinant and Inverse of 2x2 matrices	Ready to use	Violeta CIT	04/10/2017 17:56
Determinant and Inverse of 2x2 matrices	draft	Marie Nicholson	04/02/2021 15:34
Ex 5 Determinant and Inverse of 2x2 matrices	Ready to use	Violeta CIT	16/09/2018 11:11
Simon's copy of Determinant and inverse of 2x2 matrices	draft	Simon Thomas	01/04/2019 15:24
Determinant and Inverse of 2x2 matrices	draft	Xiaodan Leng	10/07/2019 23:31
Determinant and inverse of 2x2 matrices	draft	Shaheen Charlwood	19/02/2020 21:17
Inverse of a 2x2 matrix	draft	Tamsin Smith	17/09/2024 13:15

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Question statement

Do the following matrix problems.

Give any introductory information the student needs.

			Name	Type	Generated Value

a11	integer	-1
a12	integer	-5
a21	integer	-2
a22	integer	1
a	matrix	matrix([-1,-5],[-2,1])

			Name	Type	Generated Value

b11	integer	8
b12	integer	3
b21	integer	-1
b22	integer	-2
b	matrix	matrix([8,3],[-1,-2])

			Name	Type	Generated Value

c11	integer	2
c12	integer	-2
c21	integer	3
c22	integer	3
c	matrix	matrix([2,-2],[3,3])

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(-9..9 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`

-1

→ Used by:

Advice

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
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3. Gap Gap 2. Number entry
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4. Gap Gap 3. Number 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 Matrix entry
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Gap-fill

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

Let

$\mathrm{A} = \var{a},\;\; \mathrm{B} = \var{b},\;\; \mathrm{C} = \var{c}$

Calculate the determinants of these matrices:

$\det\left(\mathrm{A}\right) =$ Gap 0.

$\det\left(\mathrm{B}\right) =$ Gap 1.

$\det\left(\mathrm{C}\right) =$ Gap 2.

$\det\left(\mathrm{ABC}\right) =$ Gap 3.

Advice

Determinant of a $2 \times 2$ matrix

The determinant of a matrix $\mathrm{M} = \begin{pmatrix} a&b \\ c&d \end{pmatrix}$ is given by

$\det\left(\mathrm{M}\right) = ad-bc$

If we have two $n \times n$ matrices $M$ and $N$ , then

$\det\left(\mathrm{MN}\right) = \det\left(\mathrm{M}\right)\det\left(\mathrm{N}\right)$

And it follows that if we have a third matrix $P$ ,

$\det\left(\mathrm{MNP}\right) = \det\left(\mathrm{M}\right)\det\left(\mathrm{N}\right)\det\left(\mathrm{P}\right)$

a)

Thus for our example we have:

$\begin{align} \det\left(\mathrm{A}\right) &= \simplify[]{{a11}*{a22}-{a12}*{a21} = {det(a)}} \\ \det\left(\mathrm{B}\right) &= \simplify[]{{b11}*{b22}-{b12}*{b21} = {det(b)}} \\ \det\left(\mathrm{C}\right) &= \simplify[]{{c11}*{c22}-{c12}*{c21} = {det(c)}} \end{align}$

$\begin{align} \det\left( \mathrm{ABC} \right) &= \det(\mathrm{A}) \det(\mathrm{B}) \det(\mathrm{C}) \\ &= \simplify[]{{det(a)}*{det(b)}*{det(c)}} \\ &= \var{det(a*b*c)} \end{align}$

Inverse of a $2 \times 2$ matrix

Suppose $\mathrm{M} = \begin{pmatrix} a&b \\ c&d \end{pmatrix}$ is a $2 \times 2$ matrix and $\det\left(\mathrm{M}\right) = \Delta \neq 0$ .

Then $\mathrm{M}$ is invertible and

$\mathrm{M}^{-1} = \frac{1}{\Delta} \begin{pmatrix} d & -b\\ -c& a \end{pmatrix}=\begin{pmatrix} \frac{d}{\Delta} & -\frac{b}{\Delta}\\ -\frac{c}{\Delta}& \frac{a}{\Delta} \end{pmatrix}$

Applying this to these examples we obtain:

b)

$\simplify[fractionnumbers]{matrix:A^(-1)={inverse(a)}}$

c)

$\simplify[fractionnumbers]{matrix:B^(-1)={inverse(b)}}$

d)

$\simplify[fractionnumbers]{matrix:C^(-1)={inverse(c)}}$

Give a worked solution to the whole question.

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This question is used in the following exam:

Linear Algebra 1 by Michael Foreman in Michael's workspace.

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Checkpoint description

Christian Lawson-Perfect 8 years, 9 months ago

Michael Foreman 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

Determinant of a $2 \times 2$ matrix

a)

Inverse of a $2 \times 2$ matrix

b)

c)

d)

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Error

Determinant and inverse of 2x2 matrices

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 8 years, 9 months ago

Michael Foreman 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Determinant of a 2×22 \times 2 matrix

a)

Inverse of a 2×22 \times 2 matrix

b)

c)

d)

Generated value: `integer`

Determinant of a $2 \times 2$ matrix

Inverse of a $2 \times 2$ matrix