Matrix products and transposes (same as WBQ 1.31)

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Julia Goedecke

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4 years, 7 months ago

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Julia Goedecke 4 years, 7 months ago

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Created this as a copy of Can you multiply these matrices? (same as WBQ 1.28).

Name	Status	Author	Last Modified
Can you multiply these matrices? (same as WBQ 1.28)	Ready to use	Julia Goedecke	25/08/2021 12:57
Can you multiply matrices of these sizes?	Ready to use	Julia Goedecke	25/08/2021 12:57
Matrix products and transposes (same as WBQ 1.31)	Ready to use	Julia Goedecke	25/08/2021 12:57
Matrix products: any combination from WBQ 1.28/1.31	Ready to use	Julia Goedecke	25/08/2021 12:57
Matrix multiplication	Ready to use	Julia Goedecke	25/08/2021 12:57
copy of Matrix products and transposes (same as WBQ 1.31)	Ready to use	Julia Goedecke	25/08/2021 12:57
Matrix multiplication	Ready to use	Julia Goedecke	25/04/2022 09:56

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

Question statement

Let $A=\var{A}$ , $B=\var{B}$ , $C=\var{C}$ , $D=\var{D}$ , $E=\var{F}$ .

You will be asked to

Determine whether certain matrix products exist.
Calculate certain matrix products.
Calculate the transpose of all matrices.
Calculate some more complicated matrix products.

When you have finished the calculations, think about what you have noticed and how this links to the theory from lectures. You can discuss this part in your Feedback Sessions.

Give any introductory information the student needs.

			Name	Type	Generated Value

A	matrix	matrix([3,0],[-1,2],[1,1])
B	matrix	matrix([4,-1],[0,2])
C	matrix	matrix([1,4,2],[3,1,5])
D	matrix	Matrix of size 3×3
F	matrix	Matrix of size 3×3
firstmatrix	matrix	matrix([3,0],[-1,2],[1,1])
secondmatrix	matrix	matrix([1,4,2],[3,1,5])
secondmatrixname	string	C
firstmatrixname	string	A

			Name	Type	Generated Value

rawunresolvedAB	list	List of 12 items
unresolvedAB	string	\begin{pmatrix}3\cdot4+0\cdot0
rawunresolvedRandomProduct	list	List of 18 items
unresolvedRandomProduct	string	\begin{pmatrix}3\cdot1+0\cdot3
rawunresolvedBTAT	list	List of 12 items
unresolvedBTAT	string	\begin{pmatrix}4\cdot3+0\cdot0
BTAT	matrix	matrix([12,-4,4],[-3,5,1])
rawunresolvedCCT	list	List of 12 items
unresolvedCCT	string	\begin{pmatrix}1\cdot1+4\cdot4

			Name	Type	Generated Value

symbols	list	Nested 2×2 list
rawgeneralA	list	List of 4 items
generalA	string	\begin{pmatrix}a_{11}&a_{12}\\
rawgeneralB	list	List of 4 items
generalB	string	\begin{pmatrix}b_{11}&b_{12}\\
rawgeneralAB	list	List of 8 items
generalAB	string	\begin{pmatrix}a_\{11\}\cdot b

			Name	Type	Generated Value

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
matrix([3,0],[-1,2],[1,1])

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([3,0],[-1,2],[1,1])

This variable doesn't seem to be used anywhere.

Parts

Explore mode options

Which products exist? Gap-fill
1. Gap Choose several from a list
  Add
Compute some matrix products Gap-fill
1. Gap Gap 0. Matrix entry
  Add
2. Gap Gap 1. Matrix entry
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Transpose a matrix Gap-fill
1. Gap A transpose Matrix entry
  Add
2. Gap B transpose Matrix entry
  Add
3. Gap C transpose Matrix entry
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4. Gap D transpose Matrix entry
  Add
5. Gap E transpose Matrix entry
  Add
6. Gap products of transposes Choose several from a list
  Add
Compute further products Gap-fill
1. Gap B^TA^T Matrix entry
  Add
2. Gap Comparison Choose several from a list
  Add
3. Gap A(BC) Matrix entry
  Add
4. Gap (AB)C Matrix entry
  Add
5. Gap CC^T Matrix entry
  1. Alt Unnamed alternative
  Add

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Advice

Which product exists?

We can multiply an $m\times k$ matrix and a $k\times n$ matrix, i.e. the number of columns of the first matrix and the number of rows of the second matrix have to match. The resulting product has size $m\times n$ .

$A$ has size $3\times 2$ , $B$ has size $2\times 2$ , so $AB$ exists but $BA$ doesn't.

$C$ has size $2\times 3$ , so both $AC$ and $CA$ exist.

$D$ hase size $3\times 3$ , so $AD$ doesn't exist but $DA$ exists.

Compute some matrix products

To compute the product, we work out "row $i$ of first matrix times column $j$ of second matrix" for each entry $i,j$ of the product, for example:

$\var{generalA}\var{generalB}=\var{generalAB}.$

So we have:

$AB=\var{A}\var{B}=\var{unresolvedAB}=\var{A*B}$

and

$\var{firstmatrixname}\var{secondmatrixname}=\var{firstmatrix}\var{secondmatrix}=\var{unresolvedRandomProduct}=\var{firstmatrix*secondmatrix}$

Transpose a matrix

When we transpose a matrix, we switch rows and columns: $(A^T)_{ij}=A_{ji}$ .

$A^T=\var{transpose(A)}, \quad B^T=\var{transpose(B)}, \quad C^T=\var{transpose(C)}, \quad D^T=\var{transpose(D)}, \quad E^T=\var{transpose(F)}.$

You can see that $A^T$ has size $2\times 3$ and $B^T$ has size $2\times 2$ , so now $A^TB^T$ does not exist as the middle sizes don't match, but $B^TA^T$ exists.

Compute further products

$B^TA^T=\var{unresolvedBTAT}=\var{transpose(B)*transpose(A)}$

Notice that $B^TA^T=(AB)^T$ . This is a general rule. So since $BA$ does not exist, $(BA)^T=A^TB^T$ also does not exist.

$A(BC)=\var{A}\var{B*C}=\var{A*B*C}$ and $(AB)C=\var{A*B}\var{C}=\var{A*B*C}$ . We notice that we get the same answer: matrix multiplication is associative. The point here is to "experience" this general rule on an example.

$CC^T=\var{C}\var{transpose(C)}=\var{unresolvedCCT}=\var{C*transpose(C)}$ .

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This question is used in the following exam:

Linear algebra by Farai Nechikwira in Farai's workspace.

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Matrix products and transposes (same as WBQ 1.31)

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Julia Goedecke 4 years, 7 months ago

Julia Goedecke 4 years, 7 months ago

Julia Goedecke 4 years, 7 months ago

Error in variable testing condition

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Generated value: `matrix`

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Error

Matrix products and transposes (same as WBQ 1.31)

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Julia Goedecke 4 years, 7 months ago

Julia Goedecke 4 years, 7 months ago

Julia Goedecke 4 years, 7 months ago

Error in variable testing condition

Error

Warning

Generated value: matrix

Parts

Objectives

Penalties

Generated value: `matrix`