Find determinants and inverses of 2x2 matrices

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Simon Thomas 6 years, 2 months ago

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Name	Status	Author	Last Modified
Find determinants and inverses of 2x2 matrices	draft	Simon Thomas	07/02/2019 10:29
John's copy of Find determinants and inverses of 2x2 matrices	draft	John Steele	13/05/2019 03:59
Maria's copy of Find determinants and inverses of 2x2 matrices	draft	Maria Aneiros	23/05/2019 06:31

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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([-8,-3],[1,2])
b	matrix	matrix([6,-4],[4,5])
c	matrix	matrix([4,-7],[5,6])

			Name	Type	Generated Value

a11	integer	-8
a12	integer	-3
a21	integer	1
a22	integer	2
b11	integer	6
b12	integer	-4
b21	integer	4
b22	integer	5
c11	integer	4
c12	integer	-7
c21	integer	5
c22	integer	6
tr1	integer	2
tr2	integer	6
tr3	integer	5
tr4	integer	6

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.

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Generated value: `matrix`

matrix([-8,-3],[1,2])

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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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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.

Calculate the determinants of these matrices.

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

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

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

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

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

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Find determinants and inverses of 2x2 matrices

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Taxonomy: mathcentre

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Simon Thomas 6 years, 2 months ago

Error in variable testing condition

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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)

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Find determinants and inverses of 2x2 matrices

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 6 years, 2 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