// Numbas version: exam_results_page_options {"name": "Matrices: Multiplication (Instructional)", "metadata": {"description": "

Multiplication of square and non square matrices, Identity matrices and zero matrices

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Multiplying matrices (pre-defined sizes in answers)

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", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

An $n \\times p$ matrix $A$ can be multiplied by a $p \\times n$ matrix $B$ to form an $n \\times m$ matrix $AB=C$.

\n

The number of columns of $A$ must match the number of rows of $B$.

\n

\n

The element in the $i^{th}$ row and $j^{th}$ column of $C$ is obtained by multiplying the $i^{th}$ row of $A$ with the $j^{th}$ column of $B$.

#### We are asked to carry out matix multiplications.

\n

\n

First of all, you should always check that the multiplication is even possible. Write down the dimensions (in order) of the two matrices:

\n

\n

The number of columns in the first must match the number of rows in the second. As a bonus this will also give you the dimensions of the product matrix.

\n

The multiplication is then carried out moving across the rows of the first matrix and down the columns of the second:

\n

\n

\n

Using this techniques will give:

\n

$A_1A_2=\\var{A1}\\var{A2}$

\n

\n

$A_1 A_2=\\var{prodA}$

\n

\n

$B_1B_2=\\var{B1}\\var{B2}$

\n

$B_1 B_2=\\var{prodB}$

\n

\n

$C_1C_2=\\var{C1}\\var{C2}$

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$C_1 C_2=\\var{prodC}$

\n

", "rulesets": {}, "variables": {"n1": {"name": "n1", "group": "Matrix A", "definition": "random(2..3)", "description": "", "templateType": "anything"}, "m1": {"name": "m1", "group": "Matrix A", "definition": "random(2..3 except n1)", "description": "", "templateType": "anything"}, "A1": {"name": "A1", "group": "Matrix A", "definition": "transpose(matrix(repeat(repeat(random(0..9),n1),m1)))", "description": "", "templateType": "anything"}, "A2": {"name": "A2", "group": "Matrix A", "definition": "transpose(matrix(repeat(repeat(random(0..9),m1),n1)))", "description": "", "templateType": "anything"}, "ProdA": {"name": "ProdA", "group": "Matrix A", "definition": "{A1}{A2}", "description": "", "templateType": "anything"}, "n2": {"name": "n2", "group": "Matrix B", "definition": "random(2..3)", "description": "", "templateType": "anything"}, "m2": {"name": "m2", "group": "Matrix B", "definition": "random(2..3 except n2)", "description": "", "templateType": "anything"}, "B1": {"name": "B1", "group": "Matrix B", "definition": "transpose(matrix(repeat(repeat(random(0..9),n2),m2)))", "description": "", "templateType": "anything"}, "B2": {"name": "B2", "group": "Matrix B", "definition": "transpose(matrix(repeat(repeat(random(0..9),m2),n2)))", "description": "", "templateType": "anything"}, "ProdB": {"name": "ProdB", "group": "Matrix B", "definition": "{B1}{B2}", "description": "", "templateType": "anything"}, "n3": {"name": "n3", "group": "Matrix C", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "m3": {"name": "m3", "group": "Matrix C", "definition": "random(1..1)", "description": "", "templateType": "anything"}, "C1": {"name": "C1", "group": "Matrix C", "definition": "transpose(matrix(repeat(repeat(random(0..9),n3),m3)))", "description": "", "templateType": "anything"}, "m3a": {"name": "m3a", "group": "Matrix C", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "C2": {"name": "C2", "group": "Matrix C", "definition": "transpose(matrix(repeat(repeat(random(0..9),m3),m3a)))", "description": "", "templateType": "anything"}, "prodC": {"name": "prodC", "group": "Matrix C", "definition": "{C1}{C2}", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Matrix A", "variables": ["n1", "m1", "A1", "A2", "ProdA"]}, {"name": "Matrix B", "variables": ["n2", "m2", "B1", "B2", "ProdB"]}, {"name": "Matrix C", "variables": ["n3", "m3", "C1", "m3a", "C2", "prodC"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

#### Carry out the following multiplications:

\n

\n

$A_1A_2=\\var{A1}\\var{A2}$

\n

$A_1 A_2=$ [[0]]

\n

\n

$B_1B_2=\\var{B1}\\var{B2}$

\n

$B_1 B_2=$ [[1]]

\n

\n

$C_1C_2=\\var{C1}\\var{C2}$

\n

$C_1 C_2=$ [[2]]

\n

\n

Multiplying matrices (student-defines sizes in answers)

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

An $n \\times p$ matrix $A$ can be multiplied by a $p \\times n$ matrix $B$ to form an $n \\times m$ matrix $AB=C$.

\n

The number of columns of $A$ must match the number of rows of $B$.

\n

\n

The element in the $i^{th}$ row and $j^{th}$ column of $C$ is obtained by multiplying the $i^{th}$ row of $A$ with the $j^{th}$ column of $B$.

#### We are asked to carry out matix multiplications.

\n

\n

First of all, you should always check that the multiplication is even possible. Write down the dimensions (in order) of the two matrices:

\n

\n

The number of columns in the first must match the number of rows in the second. As a bonus this will also give you the dimensions of the product matrix.

\n

The multiplication is then carried out moving across the rows of the first matrix and down the columns of the second:

\n

\n

\n

Using this techniques will give:

\n

$A_1A_2=\\var{A1}\\var{A2}$

\n

\n

$A_1 A_2=\\var{prodA}$

\n

\n

$B_1B_2=\\var{B1}\\var{B2}$

\n

$B_1 B_2=\\var{prodB}$

\n

\n

$C_1C_2=\\var{C1}\\var{C2}$

\n

$C_1 C_2=\\var{prodC}$

\n

", "rulesets": {}, "variables": {"n1": {"name": "n1", "group": "Matrix A", "definition": "random(2..3)", "description": "", "templateType": "anything"}, "m1": {"name": "m1", "group": "Matrix A", "definition": "random(2..3 except n1)", "description": "", "templateType": "anything"}, "A1": {"name": "A1", "group": "Matrix A", "definition": "transpose(matrix(repeat(repeat(random(0..9),n1),m1)))", "description": "", "templateType": "anything"}, "A2": {"name": "A2", "group": "Matrix A", "definition": "transpose(matrix(repeat(repeat(random(0..9),m1),n1)))", "description": "", "templateType": "anything"}, "ProdA": {"name": "ProdA", "group": "Matrix A", "definition": "{A1}{A2}", "description": "", "templateType": "anything"}, "n2": {"name": "n2", "group": "Matrix B", "definition": "random(2..3)", "description": "", "templateType": "anything"}, "m2": {"name": "m2", "group": "Matrix B", "definition": "random(2..3 except n2)", "description": "", "templateType": "anything"}, "B1": {"name": "B1", "group": "Matrix B", "definition": "transpose(matrix(repeat(repeat(random(0..9),n2),m2)))", "description": "", "templateType": "anything"}, "B2": {"name": "B2", "group": "Matrix B", "definition": "transpose(matrix(repeat(repeat(random(0..9),m2),n2)))", "description": "", "templateType": "anything"}, "ProdB": {"name": "ProdB", "group": "Matrix B", "definition": "{B1}{B2}", "description": "", "templateType": "anything"}, "n3": {"name": "n3", "group": "Matrix C", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "m3": {"name": "m3", "group": "Matrix C", "definition": "random(1..1)", "description": "", "templateType": "anything"}, "C1": {"name": "C1", "group": "Matrix C", "definition": "transpose(matrix(repeat(repeat(random(0..9),n3),m3)))", "description": "", "templateType": "anything"}, "m3a": {"name": "m3a", "group": "Matrix C", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "C2": {"name": "C2", "group": "Matrix C", "definition": "transpose(matrix(repeat(repeat(random(0..9),m3),m3a)))", "description": "", "templateType": "anything"}, "prodC": {"name": "prodC", "group": "Matrix C", "definition": "{C1}{C2}", "description": "", "templateType": "anything"}, "row1A": {"name": "row1A", "group": "Matrix A", "definition": "A1[0][0..1]", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Matrix A", "variables": ["n1", "m1", "A1", "A2", "ProdA", "row1A"]}, {"name": "Matrix B", "variables": ["n2", "m2", "B1", "B2", "ProdB"]}, {"name": "Matrix C", "variables": ["n3", "m3", "C1", "m3a", "C2", "prodC"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

#### Carry out the following multiplications:

\n

You need to define the dimensions of the matrix before entering your answer.

\n

\n

$A_1A_2=\\var{A1}\\var{A2}$

\n

$A_1 A_2=$ [[0]]

\n

\n

$B_1B_2=\\var{B1}\\var{B2}$

\n

$B_1 B_2=$ [[1]]

\n

\n

$C_1C_2=\\var{C1}\\var{C2}$

\n

$C_1 C_2=$ [[2]]

\n

\n

Multiplying matrices (pre-defined sizes in answers)

\n

This set is designed to emphasise non-commutativity.

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

An $n \\times p$ matrix $A$ can be multiplied by a $p \\times n$ matrix $B$ to form an $n \\times m$ matrix $AB=C$.

\n

The number of columns of $A$ must match the number of rows of $B$.

\n

\n

The element in the $i^{th}$ row and $j^{th}$ column of $C$ is obtained by multiplying the $i^{th}$ row of $A$ with the $j^{th}$ column of $B$.

#### We are asked to carry out matix multiplications.

\n

\n

First of all, you should always check that the multiplication is even possible. Write down the dimensions (in order) of the two matrices:

\n

\n

The number of columns in the first must match the number of rows in the second. As a bonus this will also give you the dimensions of the product matrix.

\n

The multiplication is then carried out moving across the rows of the first matrix and down the columns of the second:

\n

\n

\n

Using this techniques will give:

\n

$A_1A_2=\\var{A1}\\var{A2}$

\n

\n

$A_1 A_2=\\var{prodA}$

\n

\n

$A_2 A_1=\\var{A2}\\var{A1}$

\n

$A_2 A_1=\\var{prodA2}$

\n

\n

$B_1B_2=\\var{B1}\\var{B2}$

\n

$B_1B_2=\\var{prodB}$

\n

\n

$B_2B_1=\\var{B2}\\var{B1}$

\n

$B_2B_1=\\var{prodB2}$

\n

\n

Hopefully, you can see that when the multiplication is reversed we get a different answer! That is very different to what we see in conventional arithmatic.

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With \"normal\" multiplication   $3 \\times 2 = 2 \\times 3$, we say that multiplication is commutative - the order does not matter. That is not the case with matrix multiplication.

\n

It is clear that $AB$ and $BA$ are not in general the same. In fact it is the exception that $AB = BA$. In the special case in which $AB = BA$ we say that the matrices $A$ and $B$ commute.

\n

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#### Carry out the following multiplications:

\n

\n

$A_1A_2=\\var{A1}\\var{A2}$

\n

$A_1 A_2=$ [[0]]

\n

\n

$A_2 A_1=\\var{A2}\\var{A1}$

\n

$A_2 A_1=$ [[1]]

\n

\n

$B_1B_2=\\var{B1}\\var{B2}$

\n

$B_1B_2=$ [[2]]

\n

\n

$B_2B_1=\\var{B2}\\var{B1}$

\n

$B_2B_1=$ [[3]]

\n

\n

From these results, what can you conclude about matrix multiplication in general?

\n

[[4]]

\n

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Multiplying matrices (pre-defined sizes in answers)

\n

Introduces unit/identity matrices

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

An $n \\times p$ matrix $A$ can be multiplied by a $p \\times n$ matrix $B$ to form an $n \\times m$ matrix $AB=C$.

\n

The number of columns of $A$ must match the number of rows of $B$.

\n

\n

The element in the $i^{th}$ row and $j^{th}$ column of $C$ is obtained by multiplying the $i^{th}$ row of $A$ with the $j^{th}$ column of $B$.

#### We are asked to carry out matix multiplications that have some (perhaps) surprising answers.

\n

\n

The matrix $\\var{I2}$ is called the identity matrix or unit matrix of order $2$, and is usually denoted by the symbol $I$. (Strictly we would write $I_2$, to indicate the size.)

\n

$I$ plays the same role in matrix multiplication as the number $1$ does in number multiplication.

\n

\n

Therefore:

\n

just as    $a \\times 1 = 1 \\times a = a$    for any number  $a$, so    $AI = IA = A$    for any matrix   $A$.

\n

", "rulesets": {}, "variables": {"n1": {"name": "n1", "group": "Matrix A", "definition": "2", "description": "", "templateType": "number"}, "m1": {"name": "m1", "group": "Matrix A", "definition": "2", "description": "", "templateType": "number"}, "A1": {"name": "A1", "group": "Matrix A", "definition": " transpose(matrix(repeat(repeat(random(0..9),m1),n1)))", "description": "", "templateType": "anything"}, "A2": {"name": "A2", "group": "Matrix A", "definition": "transpose(matrix(repeat(repeat(random(0..9),m1),n1)))", "description": "", "templateType": "anything"}, "ProdA": {"name": "ProdA", "group": "Matrix A", "definition": "{A1}{A2}", "description": "", "templateType": "anything"}, "n2": {"name": "n2", "group": "Matrix B", "definition": "3", "description": "", "templateType": "number"}, "m2": {"name": "m2", "group": "Matrix B", "definition": "3", "description": "", "templateType": "number"}, "B1": {"name": "B1", "group": "Matrix B", "definition": "transpose(matrix(repeat(repeat(random(0..9),n2),m2)))", "description": "", "templateType": "anything"}, "B2": {"name": "B2", "group": "Matrix B", "definition": "transpose(matrix(repeat(repeat(random(0..9),m2),n2)))", "description": "", "templateType": "anything"}, "ProdB": {"name": "ProdB", "group": "Matrix B", "definition": "{B1}{B2}", "description": "", "templateType": "anything"}, "n3": {"name": "n3", "group": "Matrix C", "definition": "3", "description": "", "templateType": "number"}, "m3": {"name": "m3", "group": "Matrix C", "definition": "3", "description": "", "templateType": "number"}, "C1": {"name": "C1", "group": "Matrix C", "definition": "transpose(matrix(repeat(repeat(random(0..9),n3),m3)))", "description": "", "templateType": "anything"}, "m3a": {"name": "m3a", "group": "Matrix C", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "C2": {"name": "C2", "group": "Matrix C", "definition": "transpose(matrix(repeat(repeat(random(0..9),m3),m3a)))", "description": "", "templateType": "anything"}, "prodC": {"name": "prodC", "group": "Matrix C", "definition": "{C1}{C2}", "description": "", "templateType": "anything"}, "row1A": {"name": "row1A", "group": "Matrix A", "definition": "A1[0][0..1]", "description": "", "templateType": "anything"}, "I2": {"name": "I2", "group": "Ungrouped variables", "definition": "id(n1)", "description": "", "templateType": "anything"}, "I3": {"name": "I3", "group": "Ungrouped variables", "definition": "id(n2)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["I2", "I3"], "variable_groups": [{"name": "Matrix A", "variables": ["n1", "m1", "A1", "A2", "ProdA", "row1A"]}, {"name": "Matrix B", "variables": ["n2", "m2", "B1", "B2", "ProdB"]}, {"name": "Matrix C", "variables": ["n3", "m3", "C1", "m3a", "C2", "prodC"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

#### Carry out the following multiplications:

\n

\n

$A_1 I_2=\\var{A1}\\var{I2}$

\n

$A_1 I_2=$ [[0]]

\n

\n

$B_1 I_3=\\var{B1}\\var{I3}$

\n

$B_1 I_3=$ [[1]]

\n

\n

$C_1 I_3=\\var{C1}\\var{I3}$

\n

$C_1 I_3=$ [[2]]

\n

\n

$I_3 C_1=\\var{C1}\\var{I3}$

\n

$I_3 C_1=$ [[3]]

Multiplying matrices (pre-defined sizes in answers)

\n

Zero matrices AND AB = 0 does not imply that either A = 0 or B = 0.

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

An $n \\times p$ matrix $A$ can be multiplied by a $p \\times n$ matrix $B$ to form an $n \\times m$ matrix $AB=C$.

\n

The number of columns of $A$ must match the number of rows of $B$.

\n

\n

The element in the $i^{th}$ row and $j^{th}$ column of $C$ is obtained by multiplying the $i^{th}$ row of $A$ with the $j^{th}$ column of $B$.

#### We are asked to carry out matix multiplications that have some (perhaps) surprising answers.

\n

\n

The matrix $\\var{Z2}$ is called the zerot matrix of order $2$, and is usually denoted by the symbol $\\underline{0}$.

\n

And:

\n

$\\Large A \\times \\underline{0}=\\underline{0} \\times A=\\underline{0}$    for any matrix $A$

\n

\n

\n

This explains why, for the first four multiplications, we get fairly predictable results:

\n

$A B=\\var{A1}\\var{Z2}$

\n

$AB=\\var{Z2}$

\n

\n

$B A=\\var{Z2}\\var{A1}$

\n

$B A=\\var{Z2}$

\n

\n

$C D=\\var{B1}\\var{Z3}$

\n

$C D=\\var{Z3}$

\n

\n

$DC=\\var{Z3}\\var{B1}$

\n

$DC=\\var{Z3}$

\n

\n

But now we get some more interesting results:

\n

$EF=\\var{X1}\\var{Y1}$

\n

$EF=\\var{Z2}$

\n

\n

$FE=\\var{Y1}\\var{X1}$

\n

$FE=\\var{Z2}$

\n

\n

In the multiplication of \"normal\" numbers

\n

$\\large ab=0$

\n

would imply that either $a=0$, or $b=0$ or both are zero. This is not necessarily true for matrices.

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#### Carry out the following multiplications:

\n

\n

$A B=\\var{A1}\\var{Z2}$

\n

$AB=$ [[0]]

\n

\n

$B A=\\var{Z2}\\var{A1}$

\n

$B A=$ [[1]]

\n

\n

$C D=\\var{B1}\\var{Z3}$

\n

$C D=$ [[2]]

\n

\n

$DC=\\var{Z3}\\var{B1}$

\n

$DC=$ [[3]]

\n

\n

$EF=\\var{X1}\\var{Y1}$

\n

$EF=$ [[4]]

\n

\n

$FE=\\var{Y1}\\var{X1}$

\n

$FE=$ [[5]]