Determinant and Inverse of 2x2 matrices

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Xiaodan Leng 5 years, 8 months ago

Created this as a copy of Ex 5 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

There are 24 other versions that do you not have access to.

Question statement

Do the following matrix problems.

Give any introductory information the student needs.

			Name	Type	Generated Value

a11	integer	-3
a12	integer	0
a21	integer	2
a22	integer	5
a	matrix	matrix([-3,0],[2,5])

			Name	Type	Generated Value

b11	integer	5
b12	integer	4
b21	integer	4
b22	integer	-8
b	matrix	matrix([5,4],[4,-8])

			Name	Type	Generated Value

c11	integer	2
c12	integer	4
c21	integer	4
c22	integer	4
c	matrix	matrix([2,4],[4,4])

			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.

Can an exam override the value of this variable?

Generated value: `integer`

-3

→ Used by:

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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3. Gap Gap 2. 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:

Apprentices Year 1 Matrices by Aoife O'Brien in Aoife's workspace.

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Checkpoint description

Xiaodan Leng 5 years, 8 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

History

Comment

Checkpoint description

Xiaodan Leng 5 years, 8 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