Quiz copy of Find determinants and inverses of 2x2 matrices

Error

There was an error loading the page.

Contributors

Marie Nicholson

History

○

Marie Nicholson 5 years, 11 months ago

Created this as a copy of Find determinants and inverses of 2x2 matrices.

Name	Status	Author	Last Modified
Find determinants and inverses of 2x2 matrices	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Mark's copy of Find determinants and inverses of 2x2 matrices	draft	Mark Hodds	07/09/2017 15:43
MATH6058 Find determinants and inverses of 2x2 matrices	draft	Catherine Palmer	18/10/2017 12:13
Find determinants and inverses of 2x2 matrices	draft	Marie Nicholson	04/02/2021 15:42
Bjørn Henrik's copy of Find determinants and inverses of 2x2 matrices	draft	Bjørn Henrik Auestad	02/12/2020 15:42
Quiz copy of Find determinants and inverses of 2x2 matrices	draft	Marie Nicholson	07/05/2021 14:34
W1b - Find determinants and inverses of 2x2 matrices	Ready to use	Timur Zaripov	17/06/2022 09:37
Priyanka's copy of Find determinants and inverses of 2x2 matrices	draft	Priyanka Pandya	10/02/2020 17:14

There are 34 other versions that do you not have access to.

Question statement

Let

$\begin{align} \mathbf{A} &= \var{a}, & \mathbf{B} &= \var{b}, & \mathbf{C} &= \var{c} \end{align}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	matrix	matrix([4,5],[-6,4])
b	matrix	matrix([7,1],[1,6])
c	matrix	matrix([2,6],[5,5])

			Name	Type	Generated Value

a11	integer	4
a12	integer	5
a21	integer	-6
a22	integer	4
b11	integer	7
b12	integer	1
b21	integer	1
b22	integer	6
c11	integer	2
c12	integer	6
c21	integer	5
c22	integer	5
tr1	integer	4
tr2	integer	7
tr3	integer	6
tr4	integer	5

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
matrix([ [a11,a12],[a21,a22] ])

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: `matrix`

matrix([4,5],[-6,4])

← Depends on:

This variable doesn't seem to be used anywhere.

Parts

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

Calculate the determinants of these matrices.

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

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

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.

Use this tab to check that this question works as expected.

There was an error which means the tests can't run:

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

This question is used in the following exam:

Quiz 5 by Marie Nicholson in Marie's linear algebra workspace.

Loading...

Error

Quiz copy of Find determinants and inverses of 2x2 matrices

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Marie Nicholson 5 years, 11 months ago

Error in variable testing condition

Error

Warning

Generated value: `matrix`

Parts

Gap-fill

Determinant of a $2 \times 2$ matrix

a)

Inverse of a $2 \times 2$ matrix

b)

c)

d)

Loading...

Error

Quiz copy of Find determinants and inverses of 2x2 matrices

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Marie Nicholson 5 years, 11 months ago

Error in variable testing condition

Error

Warning

Generated value: matrix

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: `matrix`

Determinant of a $2 \times 2$ matrix

Inverse of a $2 \times 2$ matrix