Explainer: Introduction to Matrices

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.

The concept of a “matrix” is one of the longest-running and well-developed ideas in all of mathematics. Having their origins as far back as 2 BC with Chinese mathematicians, matrices are a central object of study in mathematics and are used frequently in other disciplines such as physics and computer science. As well as the applications of matrices to other science disciplines, the properties of matrices are interesting for their own reasons and, as such, are also studied within mathematics in a more theoretical and abstract framework.

There are many ways to motivate the study of matrices, such as through systems of linear equations or through geometric transformations, but these approaches rely on first establishing a correct 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 “entry” 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 “matrix” is a rectangular array of entries that are aligned in terms of rows and columns. We can write a matrix 𝐴 as 𝐴=[𝑎𝑖𝑗], where the quantity 𝑎𝑖𝑗 is the value which appears is in the 𝑖th row and the 𝑗th column.

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=1102511.

Note that there are 2 rows and 3 columns in 𝐴 and there are 6 entries in total. We say that 𝐴=[𝑎𝑖𝑗], where the 𝑎𝑖𝑗 is 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 in orange, 𝐴=1102511, and then we highlight all of the entries in the second column in blue, 𝐴=1102511.

The entry in the first row and second column, which we denote as 𝑎12, is the only entry that has been highlighted twice, which we now show in green: 𝐴=1102511.

We would therefore say that 𝑎12=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 in orange, 𝐴=1102511, and all of the entries in the first column in blue, 𝐴=1102511,

The only entry that has been highlighted twice is the entry 𝑎21=2, which is shown below in green:𝐴=1102511.

Continuing this process gives the remaining entries 𝑎11=1, 𝑎13=0, 𝑎22=5, and 𝑎23=11.

Even mathematicians who have worked with matrices for a very long time still sometimes misremember in which order we write the “𝑖” and “𝑗” 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 “row 𝑖, column 𝑗.

Example 1: Identifying a 1 × 2 Matrix given Its Elements Order

Which of the following is the matrix 𝐴=[𝑎𝑖𝑗], where 𝑖=1 and 𝑗=1,2?

  1. [𝑎12𝑎11]
  2. 𝑎11𝑎21
  3. 𝑎12𝑎11
  4. [𝑎11𝑎12]

Answer

We first think about the number of rows. Given that 𝑖=1, we know that there is one row in matrix 𝐴, which excludes options (B) and (C), since both have two rows. We then observe that in option (A) the matrix [𝑎12𝑎11] has an entry in the top left which is labeled “𝑎12.” However, this entry appears in the first row, meaning that 𝑖=1, and the first column, meaning that 𝑗=1. Therefore, this element should actually be labeled “𝑎11,” which implies that this is not the correct matrix. The only remaining option is (D), which is the matrix [𝑎11𝑎12].

We will check that this is the correct matrix. The entry in the top left is in the first row and the first column, meaning that 𝑖=1 and 𝑗=1, which correctly corresponds to the given entry “𝑎11,” The entry in the top right is in the first row and the second column, which means that 𝑖=1 and 𝑗=2. Since this entry is labeled “𝑎12,” we have confirmed that this is indeed the correct matrix.

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.

Example 2: Identifying a 2 × 3 Matrix given Its Elements Order

Which of the following is the matrix 𝐴=[𝑎𝑖𝑗], where 𝑖=1,2 and 𝑗=1,2,3?

  1. 𝑎11𝑎12𝑎13𝑎21𝑎22𝑎23
  2. 𝑎11𝑎12𝑎21𝑎22𝑎31𝑎32
  3. 𝑎11𝑎21𝑎31𝑎12𝑎22𝑎32
  4. 𝑎11𝑎21𝑎12𝑎22𝑎13𝑎23

Answer

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 𝑎11𝑎12𝑎13𝑎21𝑎22𝑎23.

For the first row, we have 𝑖=1 and, reading left-to-right across this row, we can see that the entries are 𝑎11, 𝑎12, and 𝑎13. 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 𝑎21, 𝑎22, and 𝑎23. 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 𝑎11𝑎21𝑎31𝑎12𝑎22𝑎32.

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!

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 3: Constructing a 2 × 2 Matrix given Its Elements

Given that 𝐴 is a matrix of order 2×2, with 𝑎11=7, 𝑎12=10, 𝑎21=9, and 𝑎22=2, find the matrix 𝐴.

Answer

From the information above, the matrix 𝐴 has two rows and two columns and therefore is of the following form: .

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

The first piece of information that we are given is that 𝑎11=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: 7.

We are told next that 𝑎12=10. Given that 𝑖=1 and 𝑗=2, we are referring to an entry in the first row and the second column, giving 710.

Our next requirement is that 𝑎21=9, which means that this value must appear in the second row and the first column. Therefore, we have 7109.

The final entry is 𝑎22=2, which appears in the second row and the second column. The completed matrix is therefore 𝐴=71092.

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 4: Constructing a Matrix given the Relation between Its Elements

Given that 𝐴 is a matrix of order 3×2, where 𝑎11=0, 𝑎12=𝑎313, 𝑎21=4, 𝑎22=12𝑎11, 𝑎31=8, and 𝑎32=14𝑎21, determine 𝐴.

Answer

The matrix has three rows and two columns and hence has the following form: .

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 𝑎11=0, 𝑎21=4, and 𝑎31=8, meaning that the matrix is as follows: 048.

Next, we use the given relation 𝑎12=𝑎313. Since 𝑎31=8, we have 𝑎12=5, giving 0548.

Now we use the given relation 𝑎22=12𝑎11. We already found that 𝑎11=0, giving 𝑎22=0 and the matrix 05408.

The final given relation is 𝑎32=14𝑎21. As 𝑎21=4, we have 𝑎32=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 5: Constructing a Matrix given a General Equation for Its Elements

Find the matrix 𝐴=[𝑎𝑥𝑦], with an order of 3×3, whose elements are given by the formula 𝑎𝑥𝑦=5𝑥+4𝑦.

Answer

This matrix has 3 rows and 3 columns and therefore has the form .

We begin by calculating all entries in the first row, for which 𝑖=1. The elements in the first row are 𝑎11, 𝑎12, and 𝑎13 and they are calculated using the given formula, 𝑎𝑥𝑦=5𝑥+4𝑦.

We calculate the element in the first row and the first column: 𝑎11=5×1+4×1=9.

Then, we calculate the element in the first row and the second column: 𝑎12=5×1+4×2=13.

Now we work out the element in the first row and the third column: 𝑎13=5×1+4×3=17, allowing us to populate the first row of the matrix: 91317.

Now we focus on the second row by setting 𝑖=2 and considering the elements 𝑎21, 𝑎22, and 𝑎23.

The entry in the second row and the first column is 𝑎21=5×2+4×1=14.

The entry in the second row and the second column is 𝑎22=5×2+4×2=18.

The entry in the second row and the third column is 𝑎23=5×2+4×3=22.

Now the second row of the matrix can be completed: 91317141822.

For the third and final row, we set 𝑖=3 and take the entries 𝑎31, 𝑎32, and 𝑎33.

The entry in the third row and the first column is 𝑎31=5×3+4×1=19.

The entry in the third row and the second column is 𝑎32=5×3+4×2=23.

The entry in the third row and the third column is 𝑎33=5×3+4×3=27.

The completed matrix is therefore 91317141822192317.

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 understanding is in which order we refer to the rows and columns when specifying the matrix entries 𝑎𝑖𝑗, which is essential to grasp in order 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=𝑎𝑖𝑗, 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 “row 𝑖, column 𝑗.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.