2x2 Inverses

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Marie Nicholson	said	Has some problems	4 years, 1 month ago
Christian Lawson-Perfect	said	Ready to use	10 years, 2 months ago

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Marie Nicholson 4 years, 1 month ago

Gave some feedback: Has some problems

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Christian Lawson-Perfect 10 years, 2 months ago

Gave some feedback: Ready to use

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Christian Lawson-Perfect 10 years, 2 months ago

Saved a checkpoint:

Replaced parts with matrix entry

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Hayley Bishop 11 years, 9 months ago

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Question statement

Let

$A=\begin{pmatrix} \var{a11}&\var{a12}\\ \var{a21}&\var{a22}\\ \end{pmatrix},\;\; B=\begin{pmatrix} \var{b11}&\var{b12}\\ \var{b21}&\var{b22}\\ \end{pmatrix},\;\; C=\begin{pmatrix} \var{c11}&\var{c12}\\ \var{c21}&\var{c22}\\ \end{pmatrix}$

Find the following matrix inverses. Input all matrix entries as fractions or integers and not as decimals.

Give any introductory information the student needs.

			Name	Type	Generated Value

a21	integer	2
a22	integer	6
b22	integer	7
b21	integer	1
b1	integer	1
s2	integer	1
s1	integer	1
b12	integer	3
b11	integer	7
tr1	integer	6
c12	integer	6
c11	integer	4
tr2	integer	7
tr4	integer	4
c22	integer	4
a11	integer	6
a12	integer	3
db	integer	46
dc	integer	-8
da	integer	30
a1	integer	3
c21	integer	4
c1	integer	-6
tr3	integer	7
a	integer	3
b	integer	5
s	integer	1
u	integer	-1
t	integer	1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
s2*random(1..6)

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`

← Depends on:

→ Used by:

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Matrix entry
  Add
Part b) Gap-fill
1. Gap Matrix entry
  Add
Part c) Gap-fill
1. Gap Matrix entry
  Add

Gap-fill

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

$A^{-1} =$ Unnamed gap

Advice

a)

Determinants.

Here is the formula for determinants:

$M = \begin{pmatrix} a & b \\ c&d \end{pmatrix} \Rightarrow \mathrm{det}\left(M\right) = ad-bc$

$\mathrm{det}\left(A\right) = \simplify[]{{a11}*{a22}-{a12}*{a21} = {dA}}$
$\mathrm{det}\left(B\right) = \simplify[]{{b11}*{b22}-{b12}*{b21} = {dB}}$
$\mathrm{det}\left(C\right) = \simplify[]{{c11}*{c22}-{c12}*{c21} = {dC}}$

Determinant of a product of matrices.

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

$\mathrm{det}\left(MN\right) = \mathrm{det}\left(M\right)\mathrm{det}\left(N\right)$
And it follows that if we have a third matrix

$P$ that:

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

Thus for our example we have:

$\begin{eqnarray*}\mathrm{det}\left(ABC\right) &=& \mathrm{det}\left(A\right)\times\mathrm{det}\left(B\right)\times\mathrm{det}\left(C\right)\\ &=& \var{dA}\times \var{dB} \times \var{dC}\\ &=& \var{dA*dB*dC} \end{eqnarray*}$

Inverse of a $2 \times 2$ matrix

Suppose $M$ is a $2 \times 2$ matrix and $\mathrm{det}\left(M\right) = \Delta \neq 0$ .

Then $M$ is invertible and:

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

Applying this to these examples we obtain:

b)

$A^{-1} = \begin{pmatrix} \simplify[std]{{a22}/{dA}} &\simplify[std]{{-a12}/{dA}}\\\simplify[std]{{-a21}/{dA}}&\simplify[std]{{a11}/{dA}}\end{pmatrix}$

c)

$B^{-1} = \begin{pmatrix} \simplify[std]{{b22}/{dB}} &\simplify[std]{{-b12}/{dB}}\\\simplify[std]{{-b21}/{dB}}&\simplify[std]{{b11}/{dB}}\end{pmatrix}$

d)

$C^{-1} = \begin{pmatrix} \simplify[std]{{c22}/{dC}} &\simplify[std]{{-c12}/{dC}}\\\simplify[std]{{-c21}/{dC}}&\simplify[std]{{c11}/{dC}}\end{pmatrix}$

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Part	Test	Passed?
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Matrix entry
Matrix entry		Hasn't run yet
Gap-fill
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Matrix entry
Matrix entry		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Matrix entry
Matrix entry		Hasn't run yet

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Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Marie Nicholson 4 years, 1 month ago

Christian Lawson-Perfect 10 years, 2 months ago

Christian Lawson-Perfect 10 years, 2 months ago

Hayley Bishop 11 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: `integer`

Parts

Gap-fill

a)

Determinants.

Determinant of a product of matrices.

Inverse of a $2 \times 2$ matrix

b)

c)

d)

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2x2 Inverses

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Marie Nicholson 4 years, 1 month ago

Christian Lawson-Perfect 10 years, 2 months ago

Christian Lawson-Perfect 10 years, 2 months ago

Hayley Bishop 11 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

a)

Determinants.

Determinant of a product of matrices.

Inverse of a 2×22 \times 2 matrix

b)

c)

d)

Generated value: `integer`

Inverse of a $2 \times 2$ matrix