8.5 Systems of Equations and Matrices: Mathematics Communication
So far, you have focused on written mathematical language. In this section, you will practise understanding and taking part in spoken mathematical discussions. You will read and analyse two different dialogues: one between two students, and one between a student and a lecturer.
The goal of this section is not just to understand these dialogues, but to help you feel confident in having similar discussions yourself. You will learn how to have a discussion about mathematics, ask and answer questions, and explain your ideas clearly to someone else. This will help you use English for practical conversations about mathematics, both in class and out of class. To do this, you will need to:
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notice useful phrases and sentence structures in the dialogues,
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memorise them,
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practise saying them,
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and finally, use them in real conversations.
Each line in the dialogues is numbered to help you talk about specific parts easily. For example, if you have a question, you can say: What does the second word from the end of line 8 mean?
By working with these dialogues, you will improve your ability to discuss mathematics in English, ask and answer questions, and communicate your ideas clearly with others.
Dialogue 1: Two students
An example dialogue between two university students, Chloe and Sophie, discussing systems of equations and matrices. The two students encounter each other in a study space and then sit down together.
Select the ‘Play’ button to watch the video. The activities are based on the dialogue below the video.
| Line | Speaker | Conversation |
|---|---|---|
| (1) | Chloe: | Hi there, Sophie. Have you got a quick minute? |
| (2) | Sophie: | Sure, I don’t have anything right now. What’s on your mind? Good lecture just now, I followed pretty much everything until the last few minutes. |
| (3) | Chloe: | Same here, the matrix part at the end lost me. Actually, that's what I wanted to ask about. I'm struggling to understand why we need matrices. Can't we just solve systems of equations the way we usually do? |
| (4) | Sophie: | You definitely can solve small systems without matrices, but matrices come in handy when you have a lot of equations and variables. Imagine trying to solve ten equations with ten unknowns by hand! |
| (5) | Chloe: | That does sound like a lot of work. So, a matrix is basically a way to organise the information? |
| (6) | Sophie: | Exactly! A matrix is just a rectangular array of numbers arranged in rows and columns. For a system of equations, we put all the coefficients into a matrix, and all the constants on the right-hand side into a vector. |
| (7) | Chloe: | That’s called the augmented matrix, right? |
| (8) | Sophie: | Yeah, that's right. And what’s great is that we can perform row operations on the matrix without changing the solution of the system. Things like swapping two rows, multiplying a row by a constant, or adding one row to another. |
| (9) | Chloe: | Why do these operations not change the solution? |
| (10) | Sophie: | Think about it. Each row represents one equation in our system. If we multiply an equation by a constant, or add two equations together, that shouldn't change the solution. |
| (11) | Chloe: | Ahh you’re right, that makes sense. Are these the elementary row operations, mentioned in the notes? |
| (12) | Sophie: | Precisely, and using these we can transform the augmented matrix into a simpler form, usually something called “reduced row echelon form”. This method is called Gaussian elimination. |
| (13) | Chloe: | What's the point of getting the matrix into that form? |
| (14) | Sophie: | Once it's in that form, the matrix has an upside-down-staircase pattern, making it straightforward to read the solution by working backwards from the bottom row. |
| (15) | Chloe: | That's very neat. Besides solving equations, are there any other uses for matrices? |
| (16) | Sophie: | Sure, matrices can also represent transformations in space, like rotations and reflections. |
| (17) | Chloe: | What happens then when we multiply matrices together? Do we get a combination of transformations? |
| (18) | Sophie: | That's right. Suppose we have a matrix for a ninety-degree clockwise rotation and another for reflection across the x-axis. Multiplying them gives us a single matrix that does both: first the rotation, then the reflection. |
| (19) | Chloe: | Ahh I see. But then how do we know if the new matrix first does the rotation and then the reflection, or vice versa? |
| (20) | Sophie: | That’s a great point! The order of multiplication for matrices matters. Or you might recall our lecturer saying that matrix multiplication is not commutative. That means A times B might be different from B times A. |
| (21) | Chloe: | I think I need to try this out later with some examples. So, suppose, we do A times B, which matrix is applied first? |
| (22) | Sophie: | Well, if we do A times B, we first apply B, then A. It's like function composition. Say we have f of g of x, we first apply g, then f. |
| (23) | Chloe: | I see the connection now. Thank you very much for all your help, Sophie, that really helped. |
| (24) | Sophie: | You are welcome, Chloe, see you later. |
Activity 1: Global understanding
Read each statement below carefully. Decide whether it is True, False, or Unknown based on the information given in the dialogue. A statement is True if the dialogue confirms it, False if the dialogue contradicts it, and Unknown if the dialogue neither confirms nor contradicts it. Next, select the ‘Answer’ button below the statements.
Statement 1: Chloe and Sophie both had the same experience of the lecture they have just left.
Statement 2: Matrices are needed to solve any system of equations.
Statement 3: Gaussian elimination makes it straightforward to read off the solution to a system of equations once the matrix has been fully reduced.
Statement 4: Commutativity is not a property of the multiplication of matrices.
Statement 5: Chloe does recall the lecturer stating that matrix multiplication is non-commutative.
Answers
Statement 1 is True. Chloe's comment in (3), Same here, … followed by the matrix part at the end lost me confirms that she shares Sophie's experience of finding the latter part of the lecture difficult.
Statement 2 is False. In (4), Sophie states that You definitely can solve small systems without matrices. However, it is worth noting that as systems become larger and more complex, matrices become an increasingly efficient and in practice essential tool.
Statement 3 is True. In (14), Sophie states exactly that.
Statement 4 is True. in (20), Sophie states that matrix multiplication is not commutative. The Language Focus below will take time with this term commutative and some of its related terms.
Statement 5 is Unknown. While Sophie mentions that lecturer saying this, Chloe doesn't explicitly respond; we don't know whether Chloe recalls this or not.
Activity 2: Identifying useful language
Question 1: The opening of the conversation, reproduced below for convenience, contains a number of useful phrases for starting a conversation in English. The extract has been divided into seven sub-parts, each indicated by a Roman numeral (i)-(vii). Next, select the ‘Answer’ button below.
| Line | Speaker | Conversation |
|---|---|---|
| (1) | Chloe: | (i) Hi there, Sophie. (ii) Have you got a quick minute? |
| (2) | Sophie: | (iii) Sure, I don’t have anything right now. (iv) What’s on your mind? (v) Good lecture just now, I followed pretty much everything until the last few minutes. |
| (3) | Chloe: | (vi) Same here, the matrix part at the end lost me. (vii) Actually, that's what I wanted to ask about. |
- A statement where one speaker agrees with the other speaker before adding some detail.
- A question where one speaker asks the other speaker what they are thinking about.
- A standard, short greeting.
- A statement where one speaker comments on a shared experience that both speakers have just had.
- A sentence where a speaker brings the conversation back to the question they want to ask.
- A follow-up to the greeting where one speaker asks the other if they have any time available.
- A response to a question where the speaker states that they are available.
Answers
| Statement | Sentence in Extract |
|---|---|
| 1. A statement where one speaker agrees with the other speaker before adding some detail. | (vi) Same here, the matrix part at the end lost me. |
| 2. A question where one speaker asks the other speaker what they are thinking about. | (iv) What’s on your mind? |
| 3. A standard, short greeting. | (i) Hi there, Sophie. |
| 4. A statement where one speaker comments on a shared experience that both speakers have just had. | (v) Good lecture just now, I followed pretty much everything until the last few minutes. |
| 5. A sentence where a speaker brings the conversation back to the question they want to ask. | (vii) Actually, that's what I wanted to ask about. |
| 6. A follow-up to the greeting where one speaker asks the other if they have any time available. | (ii) Have you got a quick minute? |
| 7. A response to a question where the speaker states that they are available. | (iii) Sure, I don’t have anything right now. |
Question 2: The seven utterances below each perform the same communicative function as one of the seven sub-parts (i)–(vii) in Question 1, but use different words and phrases. First, match each utterance (1)–(7) to the sub-part (i)–(vii) it corresponds to from the original dialogue. This should then allow you to create a second conversation from the phrases below that is parallel to the original conversation. Next, select the ‘Answer’ button below.
- Do you have a second to chat at all?
- How can I help?
- I enjoyed the 09:00 seminar; most of it was clear but I got a bit lost in the last few minutes.
- As it happens, that's just what I wanted to talk to you about.
- Me too; the final problem at the end was a bit beyond me.
- Morning, Sophie.
- Absolutely. I'm free till 11:00.
Answers
| Utterance from the list above | Utterance from the original dialogue |
|---|---|
| 1. Do you have a second to chat at all? | (ii) Have you got a quick minute? |
| 2. How can I help? | (iv) What’s on your mind? |
| 3. I enjoyed the 09:00 seminar; most of it was clear but I got a bit lost in the last few minutes. | (v) Good lecture just now, I followed pretty much everything until the last few minutes. |
| 4. As it happens, that's just what I wanted to talk to you about. | (vii) Actually, that's what I wanted to ask about. |
| 5. Me too; the final problem at the end as a bit beyond me. | (vi) Same here, the matrix part at the end lost me. |
| 6. Morning, Sophie. | (i) Hi there, Sophie. |
| 7. Absolutely. I'm free till 11:00. | (iii)Sure, I don’t have anything right now. |
The utterances would then form the following conversation:
Chloe: Morning, Sophie. Do you have a second to chat at all?
Sophie: Absolutely. I'm free till 11:00. How can I help? I enjoyed the 09:00 seminar; most of it was clear but I got a bit lost in the last few minutes.
Chloe: Me too; the final problem at the end as a bit beyond me. As it happens, that's just what I wanted to talk to you about.
You may wish to practise one or both of these similar conversational opening. Alternatively, you may wish to create a further conversational opening using different phrases for the functions listed in the table.
Mathematics Focus: The prefixes non- and anti- in mathematics
This Mathematics Focus introduces three related technical terms in mathematics and examines two negative prefixes, non- and anti-, that are used to form opposites in mathematical English. Similar activities that look at prefixes or suffixes can be found in Language Focus 3 in Section 1.6 and the Mathematics Focus in Section 7.3.
Question 1a: Commutative and non-commutative
In line (20), Sophie states that matrix multiplication is not commutative. A binary operation [latex]∗[/latex] is said to be commutative if the order of the inputs does not affect the result, that is, if [latex]A * B = B * A[/latex] for all valid inputs [latex]A[/latex] and [latex]B[/latex]. The opposite is non-commutative.
To check whether an operation is commutative, we ask: does reversing the order of the inputs always give the same result?
For example, consider addition over the real numbers. Is it the case that [latex]a + b = b + a[/latex] for all real numbers [latex]a[/latex] and [latex]b[/latex]? The answer is yes, addition is commutative.
Now consider the three operations below. For each one, decide whether it is commutative or non-commutative and give a brief example to support your answer.
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multiplication
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division
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exponentiation
Answers
- Multiplication - commutative.
The order of the factors does not affect the product: [latex]5 \times 8 = 8 \times 5 = 40[/latex]. This holds more generally: in any product of several numbers, the factors may be rearranged in any order without changing the result, for example, [latex]3 \times 4 \times 5 = 4 \times 5 \times 3 = 4 \times 3 \times 5 = 60[/latex]. - Division - non-commutative.
Reversing the order of the inputs changes the result: [latex]6 \div 3 = 2[/latex], whereas [latex]3 \div 6 = \frac{1}{2}[/latex]. These are not equal, so division is non-commutative. - Exponentiation - non-commutative.
The base and the exponent play different roles: [latex]4^3 = 64[/latex], whereas [latex]3^4 = 81[/latex]. In general, [latex]a^b \neq b^a[/latex], for example, [latex]2^3 = 8[/latex] whereas [latex]3^2 = 9[/latex].
Question 1b: Anti-commutative
Now consider subtraction. We can verify that subtraction is not commutative: [latex]9 - 3 = 6[/latex], whereas [latex]3 - 9 = -6[/latex], so [latex]9 - 3 \neq 3 - 9[/latex].
However, notice that [latex]9 - 3 = -(3 - 9)[/latex]. More generally, [latex]x - y = -(y - x)[/latex] for all real numbers [latex]x[/latex] and [latex]y[/latex]. Given this relationship, is non-commutative really the most precise term for subtraction?
Answers
We could say that subtraction is non-commutative, and this would not be wrong. However, it would not be fully precise either. Because [latex]x - y = -(y - x)[/latex], reversing the order of the inputs does not give an unrelated result; it gives the negative of the original result. This is a stronger and more specific relationship than simply being non-commutative.
The correct term for this property is anti-commutative: an operation ∗ is anti-commutative if [latex]A * B = -(B * A)[/latex] for all valid inputs. Note the distinction between the two prefixes:
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non- means simply not, that is, the property does not hold.
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anti- means opposite in a specific, structured way, that is, reversing the inputs reverses the sign of the result.
Subtraction is therefore more precisely described as anti-commutative rather than simply non-commutative.
Question 2: The pattern across mathematics
The same pattern of three terms (a base term, a non- form, and an anti- form) is not unique to this operation and it appears across several areas of mathematics:
| Base term | non- form | anti- form |
|---|---|---|
| commutative | non-commutative | anti-commutative |
| symmetric | non-symmetric | anti-symmetric |
| linear | non-linear | anti-linear |
| homomorphism | non-homomorphism | anti-homomorphism |
In each case, the non- form simply means that the base property does not hold. The anti- form, by contrast, usually indicates a more structured kind of opposition or reversal, but the exact meaning depends on the mathematical context. This distinction is worth keeping in mind as you encounter these terms in your studies.
You may wish to spend some time researching one or more of these sets of terms. This will help you develop familiarity with their mathematical meanings and, where possible, the ability to give examples and apply them in calculations. It will also deepen your understanding of the distinction between non- and anti- as they are used in mathematical English, a distinction that is easy to overlook but mathematically significant.
Activity 3: Your turn!
Goal
Practise having a conversation where one student asks another for help. Use the dialogue above as a guide and Focus on communication, not necessarily on solving new math problems. Working together, create a set of confusions that you have or have had and use these to construct the dialogue.
Set-up
In pairs, take the following roles:
Speaker A is the student who has a question or set of questions about matrices. Speaker A can either choose something that is an actual problem, or something that they already understand and pretend they don't.
Speaker B is the student that has the answer to the question and needs to explain it to Speaker A.
Procedure
Step 1: Assign the roles - one student takes Speaker A and the other student takes Speaker B.
Step 2: Decide on the question or problem that the dialogue will explore. As noted above, this may be a real, actual problem that one of the two students has, or something that both students understand but pretend they do not for the purpose of the dialogue. The problem might be about terminology, concepts, methods, definitions, examples, application or something else.
Step 3: Take 3 to 5 minutes to look over the dialogue above and note down any useful phrases.
Step 4: Run the dialogue. Speaker A should start either with a variation on Chloe’s opening above, or with another suitable opening, for example: Hi [Speaker B], have you got a quick minute? I’m struggling with [Problem P].
Step 5: Switch roles and practise again to experience both asking and answering.
Step 6: Together reflect on what went well. You might reflect on how accurately and fluently the dialogue was produced, how confident you felt, and if there are alternative ways of going through the dialogue.
Dialogue 2: A student and a lecturer
An example dialogue between a university student, Mia, and her lecturer, Dr Andrew Morrison, discussing row reduction and the augmented matrix. The student approaches the lecturer, straight after a lecture.
Select the ‘Play’ button to watch the video. The activities are based on the dialogue below the video.
| Line | Speaker | Conversation |
|---|---|---|
| (1) | Mia: | Excuse me, Dr Morrison, do you have a moment? |
| (2) | Dr Morrison: | Ah, hello Mia. Yes, let's talk here. And please, just call me 'Andrew'. |
| (3) | Mia | Thank you. So, I've been working through the row reduction examples, but I'm not entirely clear on why row operations don’t change the solution. |
| (4) | Dr Morrison | That's an excellent question. First, what do the rows in our augmented matrix represent? |
| (5) | Mia: | Each row represents one of the equations of our system, right? |
| (6) | Dr Morrison: | Exactly, so if we swap two equations, we haven't changed what we're solving; we're still solving the same system. |
| (7) | Mia: | Ah, I see. So, swapping rows is just like reordering the equations? |
| (8) | Dr Morrison: | Precisely! The same logic applies to the other operations. When we multiply an equation by a non-zero constant, we haven't changed the equation’s information. For example, x plus two y equals five is the same as two x plus four y equals ten. |
| (9) | Mia: | What about adding one row to another? How does that preserve the solution? |
| (10) | Dr Morrison: | That's a bit more subtle. If you add together two equations, you create a new one. A solution of the original two equations must also satisfy this new equation, since it’s just a combination of them. So, replacing one equation with this combined equation doesn't change the solution set. |
| (11) | Mia: | Hmm, that kind of makes some sense. But how would we prove that it doesn’t add any solutions? |
| (12) | Dr Morrison: | OK, let’s be rigorous. What we want to prove is that our solution satisfies equations one and two, if and only if our solution satisfies equations one and one plus two. |
| (13) | Mia: | Agreed, so we have two directions to prove. |
| (14) | Dr Morrison: | Great, let’s start from left-to-right. Suppose our solution satisfies equations one and two. Then certainly, our solution satisfies equation one, and it also satisfies one plus two. |
| (15) | Mia: | Yes, that makes sense. How about the right-to-left direction? |
| (16) | Dr Morrison: | Suppose that our solution satisfies equations one and one plus two. Then definitely it satisfies equation one. So, it remains to show it satisfies equation two. That’s true since equation two is equal to equation one plus two, minus equation one. |
| (17) | Mia: | Ohh, I get it now. So, what is the point of these row operations, anyway? |
| (18) | Dr Morrison: | By systematically adding and eliminating rows, we transform the matrix into a form where the solution becomes immediate. That's the essence of Gaussian elimination. |
| (19) | Mia: | What if the system has no solution? |
| (20) | Dr Morrison: | If there's no solution, you'll get a contradiction, like a row saying zero equals one. |
| (21) | Mia: | What if there are infinitely many solutions? |
| (22) | Dr Morrison: | That happens when you have fewer equations than unknowns, or when an equation depends on another. Then, some variables become what we call free variables; they can take any value, and the other variables are expressed in terms of them. |
| (23) | Mia: | And this Gaussian elimination works for any size system? |
| (24) | Dr Morrison: | Yes, whether you have 3 or 300 equations, the process is the same. We can program this procedure into a computer and use it to solve huge systems. |
| (25) | Mia: | Thank you, Dr Morrison - sorry, Andrew. I feel much more confident about this now. |
| (26) | Dr Morrison: | You're very welcome, Mia. Keep practising the row operations, and they’ll soon become second nature. |
Activity 1: Global understanding
In the previous dialogue, we saw how examples, counter-examples, and non-examples play a central role in mathematical thinking and communication. These appear again in the dialogue between Mia and Dr Morrison.
Your task: Read the dialogue and decide whether each of the following statements is True, False, or Unknown. A statement is True if the dialogue confirms it, False if the dialogue contradicts it, and Unknown if the dialogue neither confirms nor contradicts it. Check your answers in the Answer section below. Next, select the ‘Answer’ button below the statements.
Statement 1: Dr Morrison prefers an informal mode of address.
Statement 2: The logic and level of difficulty of Dr Morrison's explanation of swapping rows is identical to his explanation of adding one row to another.
Statement 3: The discussion between Mia and Dr Morrison covers four possible outcomes of applying Gaussian elimination.
Statement 4: Mia finds the mathematics of larger systems more interesting than that of smaller ones.
Answers
Statement 1 is True. Dr Morrison invites Mia to address him by his first name, which she then does. It is worth noting that addressing a member of academic staff by their first name is common in many contemporary UK university contexts, but may be considered more or less appropriate in other cultural settings. If you are unsure about the conventions at your own institution, it is always safer to use a title and surname until invited to do otherwise.
Statement 2 is False. In lines (6)-(8), Dr Morrison compares the row-swapping operation to multiplying an equation by a non-zero constant, suggesting that it is a relatively straightforward operation. By contrast, in line (10), he describes the operation of adding one row to another as more subtle and provides a more detailed explanation. His explanations strongly suggest that the two operations are not treated as identical in either their logic or their level of difficulty.
Statement 3 is False. The discussion covers three possible outcomes, not four. When Gaussian elimination is applied to a system of equations, exactly three outcomes are possible: the system has a unique solution, the system has no solution (in which case it is said to be inconsistent), or the system has infinitely many solutions, expressed using one or more arbitrary parameters in the general solution.
Statement 4 is Unknown. Although the dialogue covers Gaussian elimination applied to systems of different sizes, Mia does not express any personal preference or opinion about the relative interest of larger versus smaller systems. Her questions are focused on understanding the mathematics rather than on evaluating it.
Activity 2: Identifying useful language
When explaining mathematics in spoken or written English, each sentence in an explanation typically performs a distinct function. Recognising these functions helps you both to understand explanations more effectively and to construct your own.
In line (10) of the dialogue, Dr Morrison gives a four-sentence explanation in response to Mia's question. The explanation is reproduced below with a commentary on the function of each sentence. Study the annotations carefully before moving on to the task below.
| Sentence of Dr Morrison's explanation in (10) | Commentary |
|---|---|
| That's a bit more subtle. | Dr Morrison signals to Mia that the answer to her question is more complex than his previous answer. This sentence prepares the listener for a more detailed explanation and sets expectations appropriately. |
| If you add together two equations, you create a new one. | Dr Morrison begins his argument with a simple, familiar statement. By starting from something Mia already knows, he builds a foundation for the more complex point that follows. |
| A solution of the original two equations must also satisfy this new equation, since it’s just a combination of them. | Dr Morrison connects the solution of the original equations to the new combined equation, establishing that the two are related. The word since signals that what follows is a justification or reason. |
| So, replacing one equation with this combined equation doesn't change the solution set. | Dr Morrison states his conclusion. The word so signals that this sentence follows logically from what has been established in the previous sentences. |
Your task: Now read Dr Morrison's response in line (16), reproduced below. For each sentence, write a commentary explaining the function it performs in the overall explanation. Next, select the ‘Answer’ button below.
Answer
| Sentence of Dr Morrison's explanation in (16) | Commentary |
|---|---|
| Suppose that our solution satisfies equations one and one plus two. | Dr Morrison begins by stating his starting assumption; that the solution satisfies both equations currently in the system. The word suppose is a common mathematical signal that a hypothesis is being introduced. |
| Then definitely it satisfies equation one. | Dr Morrison draws an immediate and straightforward consequence from the assumption stated in the previous sentence. The word then signals that this follows directly. The word definitely emphasises that no further justification is needed; the result is obvious from the hypothesis. |
| So, it remains to show it satisfies equation two. | Dr Morrison narrows the argument to the one thing that still needs to be established. The phrase it remains to show is a common and useful expression in mathematical writing and speech, signalling that the argument is nearly complete and only one step is left. |
| That’s true since equation two is equal to equation one plus two, minus equation one. | Dr Morrison completes the argument by providing the justification for the final step. The word since again signals a reason or justification. |
Activity 3: Your turn!
Goal
Practise having conversations about mathematics problems about systems of equations and matrices in English. Focus on communication, not necessarily on solving new math problems.
Set-up
In pairs, take the following roles:
Speaker A is the student with a question about the mathematics of systems of equations and matrices. Speaker A can either choose something that is an actual problem, or something that they already understand and pretend they don’t.
Speaker B is the lecturer, professor or seminar leader with the answer to the question and needs to explain it to Speaker A.
Procedure
Step 1: Assign the roles - one student takes Speaker A and the other student takes Speaker B. Think about how this student-lecturer dialogue might be different to the student-student dialogue in the beginning of the section. The lecturer is highly likely to know the answers to the problems but might also ask helpful questions or provide partial examples in order to guide the student.
Step 2: Decide on a problem from the topic of systems of equations and matrices to discuss.
Step 3: Take 3 to 5 minutes to look over the dialogue above and note down any phrases that you feel may be useful.
Step 4: Run the dialogue. Speaker A should start, either with a variation on Mia's opening utterance above, or with something else, for example, Excuse me Dr [Name], would you happen to have some time to help me out with [Problem P]. It shouldn't take too long.
Step 5: Run the dialogue a second time with the roles reversed.
Step 6: Together reflect on what went well. You might reflect on how accurately and fluently the dialogue was produced; how confident you felt; and if there are alternative ways of going through the dialogue.
In this section, you practised reading and performing dialogues (spoken communication) about mathematics. In the next section, you will focus on written mathematical communication.
