Lesson Explainer: Introduction to Matrices Mathematics

In this explainer, we will learn how to identify matrices and determine the order of a matrix and the position of each of its elements.

In mathematics, the idea of using an array of numbers to solve problems has existed as far back as 200 BC, when Chinese mathematicians used them to solve systems of linear equations. Despite existing for a long time, it was only in 1850 that this concept was formalized by the term matrix (plural matrices), by James Joseph Sylvester. Since then, matrices have become a central object of study in mathematics and are frequently used in other disciplines such as physics and computer science. As one example, 3D graphics are highly dependent on matrices, as they can be used to represent transformations of points in space.

Let us begin by establishing, in fundamental terms, an understanding of what matrices are. A matrix is a rectangular array which is separated into rows (which run horizontally) and columns (which run vertically). Each β€œentry” in a matrix is then described with respect to the row and column that it appears in, which is defined as follows.

Definition: Matrix Entries

A β€œmatrix” is a rectangular array of numbers that are aligned in terms of rows and columns. We can write a matrix 𝐴 as 𝐴=(π‘Ž), where the quantity π‘Žοƒο… is the value which appears is in the 𝑖th row and the 𝑗th column.

We refer to the π‘Žοƒο… as the elements (or entries) of the matrix.

It is easier to demonstrate this concept than it is to describe it, so we will begin with an example. We first define the matrix 𝐴=ο€Ό1βˆ’1025βˆ’11.

Note that there are 2 rows and 3 columns in 𝐴 and there are 6 entries in total. We say that 𝐴=(π‘Ž), where the π‘Žοƒο… are the entries of this matrix as described in the definition above. Since there are 2 rows, we have 𝑖=1,2. Similarly, there are 3 columns and hence 𝑗=1,2,3. As an example, if we set 𝑖=1 and 𝑗=2 then we are looking in the first row and the second column. To locate this entry, initially we highlight all of the entries in the first row,

and then we highlight all of the entries in the second column,

The entry in the first row and second column, which we denote as π‘ŽοŠ§οŠ¨, is the only entry that has been highlighted twice, which we show below:

We would therefore say that π‘Ž=βˆ’1.

If we were then to set 𝑖=2 and 𝑗=1, we would be referring to the entry in the second row and the first column. Repeating the same process as above, we now highlight all entries in the second row,

and all of the entries in the first column,

The only entry that has been highlighted twice is the entry π‘Ž=2, which is shown below:

Continuing this process gives the remaining entries π‘Ž=1, π‘Ž=0, π‘Ž=5, and π‘Ž=βˆ’11.

In the following example, we will practice using this method to identify specific elements of a matrix using their indices.

Example 1: Finding an Element of a Given Matrix

Given that 𝐴=ο€Όβˆ’24βˆ’7βˆ’199,𝐡=ο€βˆ’732βˆ’6βˆ’4βˆ’8730,𝐢=2βˆ’5βˆ’4, find π‘ŽοŠ¨οŠ©, π‘οŠ¨οŠ§, and π‘οŠ¨οŠ§.

Answer

Recall that when we refer to element π‘Žοƒο… of a matrix 𝐴, we mean the entry in row 𝑖 and column 𝑗 of that matrix. Since we have to find π‘ŽοŠ¨οŠ©, π‘οŠ¨οŠ§, and π‘οŠ¨οŠ§, this means we have to find one entry from each of the three matrices 𝐴, 𝐡, and 𝐢.

Let us begin by considering π‘ŽοŠ¨οŠ©. As 𝑖=2 here, this refers to an entry in row 2 of 𝐴, which we highlight below:

As 𝑗=3, this entry is also in column 3, which we highlight below:

Therefore, π‘ŽοŠ¨οŠ© is the entry in the second row and third column, which is the only entry highlighted twice, as shown below:

Thus, π‘Ž=9.

We can continue this process for the other two matrices. For π‘οŠ¨οŠ§, 𝑖=2, so let us highlight row 2 of 𝐡:

The value of 𝑗=1, which means column 1, which we highlight below:

Combining the rows and columns together, we see that π‘οŠ¨οŠ§ is the only entry highlighted twice:

Therefore, 𝑏=βˆ’6.

For the final matrix, we have to find π‘οŠ¨οŠ§. Here, 𝑖=2 refers to the second row of 𝐢. We highlight this row below:

Now, for the column, we have 𝑗=1. However, in this case matrix 𝐢 only has one column, so there is no need to highlight it. We can conclude that the entry 𝑐=βˆ’5.

To summarize the results, we have π‘Ž=9, 𝑏=βˆ’6, and 𝑐=βˆ’5.

Even mathematicians who have worked with matrices for a very long time still sometimes misremember in which order we write the β€œπ‘–β€ and β€œπ‘—β€ when referring to the rows and columns. We will practice this very shortly but a reasonably good starting point is to remember that we generally refer to the row before we refer to the column. Some people simply prefer to remember the phrase β€œrow 𝑖, column 𝑗.”

This ordering of the rows first and the columns second also applies when we refer to the dimensions of a matrix, which we can do as follows.

Definition: Order of a Matrix

If a matrix has π‘š rows and 𝑛 columns, it is said to have orderΒ π‘šΓ—π‘› (read β€œπ‘š by 𝑛”).

So, for instance, 𝐴=4βˆ’330βˆ’76 is a matrix with order 3Γ—2 (three by two), since it has three rows and two columns. Alternatively, we can omit the word β€œorder” and just refer to it as a 3Γ—2 matrix. Additionally, let us note that the number of elements (or entries) in 𝐴 is 6, which is equal to the number of rows (3) times the number of columns (2). In general, since matrices are rectangular, this leads to the following rule that we can use to find the number of elements in a matrix.

Rule: Number of Elements in a Matrix

An π‘šΓ—π‘› matrix has π‘šΓ—π‘› elements.

This rule makes it very simple to find the number of elements in a matrix, provided we are given the order, as we will see in the following example.

Example 2: Finding the Number of Elements in a Matrix given Its Order

How many elements are there in a matrix of order 9Γ—7?

Answer

Recall that the notation 9Γ—7 refers to a matrix with 9 rows and 7 columns. If we wanted to manually find out how many elements this matrix has, we could draw out an arbitrary matrix with 9 rows and 7 columns and count every single element.

An easier approach is to realize that because matrices are rectangular, the number of elements is equal to the number of rows times the number of columns. That is, 9Γ—7=63.

Visually, we can see this below.

Thus, the number of elements is 63.

In our next example, we will practice using the π‘Žοƒο… notation to refer to the entries of a matrix.

Example 3: Identifying a 2 Γ— 3 Matrix given Its Elements Order

Which of the following is the matrix 𝐴=(π‘Ž), where 𝑖=1,2 and 𝑗=1,2,3?

  1. ο€Όπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘ŽοˆοŠ§οŠ§οŠ§οŠ¨οŠ§οŠ©οŠ¨οŠ§οŠ¨οŠ¨οŠ¨οŠ©
  2. οπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘ŽοοŠ§οŠ§οŠ§οŠ¨οŠ¨οŠ§οŠ¨οŠ¨οŠ©οŠ§οŠ©οŠ¨
  3. ο€Όπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘ŽοˆοŠ§οŠ§οŠ¨οŠ§οŠ©οŠ§οŠ§οŠ¨οŠ¨οŠ¨οŠ©οŠ¨
  4. οπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘ŽοοŠ§οŠ§οŠ¨οŠ§οŠ§οŠ¨οŠ¨οŠ¨οŠ§οŠ©οŠ¨οŠ©

Answer

Let us remind ourselves that π‘Žοƒο… refers to the entry of the matrix in row 𝑖, column 𝑗 and that the combination of these entries into rows and columns forms the matrix 𝐴=(π‘Ž).

Since 𝑖=1,2 and 𝑗=1,2,3, we know that there are two rows and three columns in matrix 𝐴. We can therefore eliminate options (B) and (D), since both have three rows and two columns. The matrix in option (A) has the form ο€Όπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žοˆ.

For the first row, we have 𝑖=1 and, reading left-to-right across this row, we can see that the entries are π‘ŽοŠ§οŠ§, π‘ŽοŠ§οŠ¨, and π‘ŽοŠ§οŠ©. The values of the 𝑖 components are all 1 and the values of the 𝑗 components increase from 1 to 3. Therefore, the first row is correct. For the second row, we observe that 𝑖=2, which is correct, and that the entries are π‘ŽοŠ¨οŠ§, π‘ŽοŠ¨οŠ¨, and π‘ŽοŠ¨οŠ©. By the same reasoning, we know that this row is also correct and hence (A) is the right answer.

We have already found the correct answer, but it is helpful to understand why the matrix in (C) is incorrect. The form of the matrix is ο€Όπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žοˆ.

Although there are two rows and three columns in this matrix, the references to the rows and columns have been switched. If we look only at the first row and read from left to right, we can see that the value of the 𝑖 component changes, rather than the values of 𝑗 components, which would suggest that we are changing rows and not columns. However, we were only looking at the first row, so this cannot be the case.

As we read from left to right across the same row, the value of the 𝑖 component should stay the same and the value of the 𝑗 component should increase. Similarly, in reading from top to bottom down the same column, the 𝑖 component should increase and the 𝑗 component should stay the same. We can use this understanding to quickly check whether or not we have the correct form of a matrix.

After a small amount of practice, it will become very natural to quickly read the entries of any given matrix, with key definitions and theorems requiring this idea to be understood intuitively. One of the best ways to practice this is with matrices that have numbers in their entries, rather than abstract symbols or unspecified variables.

Example 4: Constructing a 2 Γ— 2 Matrix given Its Elements

Given that 𝐴 is a matrix of order 2Γ—2, with π‘Ž=7, π‘Ž=βˆ’10, π‘Ž=βˆ’9, and π‘Ž=2, find the matrix 𝐴.

Answer

Recall that the notation 2Γ—2 means that the matrix 𝐴 has two rows and two columns and therefore is of the following form: ο€»βˆ—βˆ—βˆ—βˆ—ο‡.

Please note that the βˆ— symbol indicates that this entry contains some unknown information and we are not assuming that all of these entries have the same value.

We recall that π‘Žοƒο… refers to the entry of the matrix in row 𝑖, column 𝑗.

The first piece of information that we are given is that π‘Ž=7. Since 𝑖=1 and 𝑗=1, we are referring to the entry in the first row and the first column. Therefore, the matrix has the following structure: ο€»7βˆ—βˆ—βˆ—ο‡.

We are told next that π‘Ž=βˆ’10. Given that 𝑖=1 and 𝑗=2, we are referring to an entry in the first row and the second column, giving ο€»7βˆ’10βˆ—βˆ—ο‡.

Our next requirement is that π‘Ž=βˆ’9, which means that this value must appear in the second row and the first column. Therefore, we have ο€Ό7βˆ’10βˆ’9βˆ—οˆ.

The final entry is π‘Ž=2, which appears in the second row and the second column. The completed matrix is therefore 𝐴=ο€Ό7βˆ’10βˆ’92.

When working with the applications of matrices, it will sometimes be the case that algebraic relationships exist between the entries. Very often these relationships will arise from physical or logistic constraints, which will subsequently ensure that the matrix in question has special properties that can be understood and made use of. Being able to understand relationships between the entries of a matrix is an essential part of gaining fluency with matrix calculations and being able to understand the deeper matrix algebra that will arise at the higher levels of study.

Example 5: Constructing a Matrix given the Relation between Its Elements

Given that 𝐴 is a matrix of order 3Γ—2, where π‘Ž=0, π‘Ž=π‘Žβˆ’3, π‘Ž=4, π‘Ž=12π‘ŽοŠ¨οŠ¨οŠ§οŠ§, π‘Ž=8, and π‘Ž=14π‘ŽοŠ©οŠ¨οŠ¨οŠ§, determine 𝐴.

Answer

Recall that a 3Γ—2 matrix has three rows and two columns and hence has the following form: ο€Ώβˆ—βˆ—βˆ—βˆ—βˆ—βˆ—ο‹.

The π‘Žοƒο… variables listed in the question are the entries of the matrix in row 𝑖, column 𝑗.

The easiest way to begin answering this question is by using the entries which have an explicit numerical value, rather than those which are specified in relation to other entries. These are the entries π‘Ž=0, π‘Ž=4, and π‘Ž=8, meaning that the matrix is as follows: 0βˆ—4βˆ—8βˆ—οŒ.

Next, we use the given relation π‘Ž=π‘Žβˆ’3. Since π‘Ž=8, we have π‘Ž=5, giving 054βˆ—8βˆ—οŒ.

Now we use the given relation π‘Ž=12π‘ŽοŠ¨οŠ¨οŠ§οŠ§. We already found that π‘Ž=0, giving π‘Ž=0 and the matrix 05408βˆ—οŒ.

The final given relation is π‘Ž=14π‘ŽοŠ©οŠ¨οŠ¨οŠ§. As π‘Ž=4, we have π‘Ž=1. The matrix is therefore 𝐴=054081.

The entries of a matrix can also be specified very rigidly, by referring to a formula which is a function of the row and column. Potentially, these functions could be complicated, but to demonstrate the concept we will only use a very simple example.

Example 6: Constructing a Matrix given a General Equation for Its Elements

Find the matrix 𝐴=(π‘Ž)ο—ο˜, with an order of 3Γ—3, whose elements are given by the formula π‘Ž=5π‘₯+4π‘¦ο—ο˜.

Answer

A 3Γ—3 matrix has 3 rows and 3 columns and therefore has the form ο€Ώβˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—ο‹.

We begin by calculating all entries in the first row, for which π‘₯=1. The elements in the first row are π‘ŽοŠ§οŠ§, π‘ŽοŠ§οŠ¨, and π‘ŽοŠ§οŠ© and they are calculated using the given formula, π‘Ž=5π‘₯+4π‘¦ο—ο˜, where π‘Žο—ο˜ refers to the entry in row π‘₯ and column 𝑦.

We calculate the element in the first row and the first column: π‘Ž=5Γ—1+4Γ—1=9.

Then, we calculate the element in the first row and the second column: π‘Ž=5Γ—1+4Γ—2=13.

Now we work out the element in the first row and the third column: π‘Ž=5Γ—1+4Γ—3=17, allowing us to populate the first row of the matrix: ο€Ώ91317βˆ—βˆ—βˆ—βˆ—βˆ—βˆ—ο‹.

Now we focus on the second row by setting π‘₯=2 and considering the elements π‘ŽοŠ¨οŠ§, π‘ŽοŠ¨οŠ¨, and π‘ŽοŠ¨οŠ©.

The entry in the second row and the first column is π‘Ž=5Γ—2+4Γ—1=14.

The entry in the second row and the second column is π‘Ž=5Γ—2+4Γ—2=18.

The entry in the second row and the third column is π‘Ž=5Γ—2+4Γ—3=22.

Now the second row of the matrix can be completed: 91317141822βˆ—βˆ—βˆ—οŒ.

For the third and final row, we set π‘₯=3 and take the entries π‘ŽοŠ©οŠ§, π‘ŽοŠ©οŠ¨, and π‘ŽοŠ©οŠ©.

The entry in the third row and the first column is π‘Ž=5Γ—3+4Γ—1=19.

The entry in the third row and the second column is π‘Ž=5Γ—3+4Γ—2=23.

The entry in the third row and the third column is π‘Ž=5Γ—3+4Γ—3=27.

The completed matrix is therefore 91317141822192327.

For our final example, we will consider a real-life instance of constructing a matrix to hold data.

Example 7: Determining the Matrix That Represents a Given Set of Data

The table below represents the prices of some drinks in a cafΓ©. The cafΓ© owner changes the prices of the drinks, so that each drink is now two times its original price. Determine the matrix that represents the new prices of the drinks.

DrinkSmall SizeBig Size
Strawberry Juice1.55.5
Orange Juice28.5
Mango Juice3.59

Answer

We first of all need to begin by constructing the matrix that represents the prices in the table. Since the table already has the prices in rows and columns, this is as straightforward as copying that data into a matrix, 𝐴. This gives us 𝐴=1.55.528.53.59.

Just as it does in the table, each row of this matrix represents a different drink (row one is strawberry juice, row two is orange juice, and row three is mango juice), and each column represents the size (column one is small size, and column two is big size).

Additionally, we recall that each entry π‘Žοƒο… in a matrix refers to the entry in row 𝑖 and column 𝑗 of that matrix. Thus, in this example, each entry π‘Žοƒο… represents the price of the drink in row 𝑖 and its size in column 𝑗. As an example, π‘Ž=9 shows us that a big mango juice costs 9.

Now, the question asks us to find the matrix where the price of each drink is two times its original price. As each entry in the matrix represents the price of a drink, this just means we have to multiply each entry in the matrix by 2. For instance, for entry π‘ŽοŠ§οŠ¨, we have 2π‘Ž=2Γ—5.5=11.

We do this for every entry in the matrix, giving us a new matrix, 𝐡: 𝐡=2Γ—1.52Γ—5.52Γ—22Γ—8.52Γ—3.52Γ—9=311417718.

This matrix representing the new prices of the drinks is our solution.

In this explainer, we have understood how to read the entries of a matrix and also how these entries may (or may not!) relate to each other. The most important concept to understand is the order in which we refer to the rows and columns when specifying the matrix entries π‘Žοƒο… to allow us to complete some of the proofs that appear in linear algebra.

Key Points

The key points from this explainer are as follows:

  • A matrix consists of rows and columns which contain entries. These entries may be numbers, variables, or functions.
  • For a matrix 𝐴=π‘Žοƒο…, the index 𝑖 refers to the row number and the index 𝑗 refers to the column number.
  • We usually specify the row number before specifying the column number. It may be helpful to remember the phrase β€œrow 𝑖, column 𝑗.”
  • If a matrix has π‘š rows and 𝑛 columns, it is said to have orderΒ π‘šΓ—π‘›, and it has π‘šΓ—π‘› elements.
  • Matrices can be used to represent real-life data.

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