In this explainer, we will learn how to interpret augmented matrices and represent systems of linear equations as an augmented matrix.
One of the oldest general problems in mathematics is to be able to solve a system of linear equations in multiple variables. The simplest nontrivial example of this would be a system of two linear equations in two variables, such as the following:
In this case the and are the variables to be found, with the numbers multiplying these being referred to as the “coefficients.” In this case the coefficient of the -term in the first equation is the number 1, and the coefficient of the -term is 3. For the second equation, the coefficient of the -term is 2 and the coefficient of the -term is .
An example with two linear equations and two variables, such as the one above, is often referred to simply as “simultaneous equations,” or more generally a system of two equations, and students often learn to solve these types of problems in their early teens. The techniques used to solve these relatively simple systems are perfectly valid, although they become much messier when extending to systems of linear equations with a greater number of variables or equations. For example, the following system has three linear equations in three variables:
This system of linear equations would require many more steps to solve than the first example, in likelyhood leading to an increased chance of mistakes being made when completing the calculations. In order to help mitigate this possibility, as well as providing a much neater environment to solve systems of linear equations, a particular method that is normally used is called “row reduction” or “Gauss–Jordan elimination.” This method aims to remove all extraneous detail by first placing all coefficients from a system of linear equations within a matrix, with each entry of this matrix corresponding to exactly one coefficient.
Consider the system of linear equations as follows:
We should ask ourselves whether or not this level of detail is needed. In other words, how can we remove any extra detail without losing any information about the above system? The answer is to first standardize the order in which the variables appear within the linear equations that are given. In this case, reading left to right, each equation helpfully begins with the -term being shown first along with the coefficient, and then the -term being displayed with this coefficient, both being on the left-hand side of the equals sign.
Suppose that we colored the the -terms in pink, and the -terms in blue. Then, the system of linear equations would be represented as
The alignment is such that the -terms (and their coefficients) appear directly underneath each other, and the same is true of the -terms. Given that this is the case, and providing that we respect this alignment, there is nothing to prevent us from writing the coefficients in the matrix
Given that the left column relates to the coefficients of the -terms and that the right column relates to the coefficients of the -terms, this matrix encodes the same information as the original system of linear equations in equation (1), only with less detail being required.
Suppose now that we wished to include all information about the system of equations given in equation (1). The terms on the right-hand side of the equal sign are also aligned, and we can represent this in the matrix where the vertical bar represents the equals signs. Thus the original system of linear equations in equation (1) has been encapsulated faithfully, but without the need for writing out the - and -terms.
For a system of linear equations such as the one above, this approach might appear excessive. However, this approach becomes more valuable when there is an increase either in the number of equations or the number of variables. So helpful is this approach that there is a special name is given to the two matrices that encode this information, as covered in the following definition.
Definition: The Coefficient and Augmented Matrices of a System of Linear Equations
Consider a general system of linear equations in the variables and the coefficients :
Then, the “coefficient matrix” is and the “augmented matrix” is
This general definition should give a better demonstration of how all salient information from a system of linear equations can be contained within the augmented matrix, thereby removing the need to constantly write out all variables that are part of the system. This supports the completion of Gauss–Jordan elimination which is the technique most commonly used to find the solution.
Using the augmented matrix contains all information about a given system of linear equations. As a slight aside, there are equivalent descriptions of such systems that can be useful in their own right, especially when they allow the understanding of the problem in terms of the standard operations of linear algebra. For example, suppose we considered the system of linear equations as used in the definition above, and we labeled the “coefficient matrix” as .
Also, suppose that we defined the “variable matrix” and the “constant matrix,” respectively, as
Using the properties of matrix multiplication, we can see that the system of linear equations can then be expressed by the simple algebraic equation . The overall goal would then be to find the variable matrix , which would give the values of all and hence solve the system of linear equations. Under the assumption that is a square matrix (which means that ), it may be the case that an inverse matrix exists such that , where is the identity matrix. If this matrix does exist, then we can find by multiplying both sides of on the left-hand side by the inverse , hence giving . By recognizing that and that , we find that this extra step of working returns the required solution . Thus, finding is reliant on finding the inverse of the coefficient matrix and then multiplying this by the constant matrix.
The above tangent demonstrates how it can be useful to contrive equivalent descriptions of systems of linear equations. The above method, however, is only useful when the system of linear equations has the same number of equations as variables, which means that the coefficient matrix is a square matrix and therefore has a well-defined notion of a multiplicative inverse (which is not the same as saying that this inverse always exists, which it does not necessarily). Often, expressing a system of linear equations in terms of the augmented matrix is the most useful starting point, as this contains all information about the coefficients and the constant terms and provides a natural arena for row operations to be completed as part of Gauss–Jordan elimination.
Note that if a system of linear equations features equations with variables, then the coefficient matrix will contain rows and columns, hence having “dimension” . The augmented matrix includes an extra column for the terms of each equation that appear on the right-hand side of the equals sign. Therefore the augmented matrix has rows and columns, and hence we would say that this has dimension .
This above definition is a very general definition that we will illustrate shortly with several examples. It should be reiterated that the applicability of this technique is entirely reliant on the variables being written in the same order within each linear equation, before the coefficients are extracted and placed into the augmented matrix. In the first few examples that we give, the equations have already been arranged in this manner. However, in later examples, this step will not have been completed and therefore we will need to be diligent in ensuring that this is the case before either the coefficient or the augmented matrices are created.
Example 1: Finding the Augmented Matrix for a System of Two Linear Equations in Two Variables
Find the augmented matrix for the following system of equations:
We begin by writing out the system of linear equations but with the -terms highlighted in pink and the -terms highlighted in blue:
As we can see, there are two equations in two variables and these equations have already been ordered so that the -terms appear first, followed by the -terms, and then the equals sign of each equation. This means that the augmented matrix has two rows and three columns, and hence has the following form:
To populate the entries of this matrix, we first examine the coefficients of the -terms in equation (2). For the first equation, the coefficient of the -term is 1, and in the second equation the coefficient of the -term is 3. Therefore, the left column of the matrix is populated with these coefficients, giving
The middle column is populated with the entries for the -coefficients in the first and second equations of (2). These coefficients both have a value of 5, meaning that the augmented matrix must have the form
The final two entries are populated with the values that appear on the right-hand side of the equals signs in both equations of (2). These are 3 and 1, respectively, and hence the completed augmented matrix is
In the previous example, the system of linear equations was written in a convenient form, with the first term of each equation being the -term, followed by the -term, and then with an equals sign immediately following this. Given that this was already the case, writing out the corresponding augmented matrix was a fairly straightforward exercise that did not require the number of steps that we gave (except for the sake of demonstration). If we are presented with the augmented matrix and asked to write out a corresponding system of linear equations, then this is usually a trivial task, as the augmented matrix will already have achieved the strict formatting regime that we require.
Example 2: Finding the System of Linear Equations Represented by 2 × 3 Augmented Matrix
Find the system of equations from the following augmented matrix:
We begin by assuming that the variables of this system are and , with corresponding to the first column of the augmented matrix and corresponding to the second column of the augmented matrix. With this being the case, the system of linear equations must take the form where the entries will be populated using the augmented matrix. The first column of the augmented matrix features the coefficients 7 and , and these are the coefficients of the -terms in the above system of linear equations. Therefore, this becomes
The middle column of the augmented matrix contains the values 2 and 4, which become the coefficients of the -terms for the first and second equation respectively. This gives the system of equations
Finally, we use the right-most column of the augmented matrix to populate the remaining blank entries of the linear system. For the first and second row, these are, respectively, and 6, and hence the complete set of linear equations is given by
Since the example above did not state specifically that we should have used the variables and , we just as easily could have used any other variables. For example, instead of and , we could have used and , which would have given
In the two examples completed above, we have only worked with augmented matrices where all entries are nonzero. This most certainly does not always have to be the case, and it is perfectly possible that we will come across augmented matrices wherein at least one entry has a value of 0. This should not cause alarm, and the treatment of such problems is essentially identical to those where all coefficients are nonzero. In the following example, we demonstrate this, giving a few extra steps to fully illustrate how to properly work with such questions.
Example 3: Finding the Augmented Matrix for a System of Three Linear Equations in Three Variables
Find the augmented matrix for the following system of equations:
The above system has three linear equations in three variables. We will begin by coloring each of these variables uniquely, as follows:
As we can see, the -terms all appear as the first terms in each equation (with the second equation featuring an -term with a coefficient of 0, which we have not included). The second terms of each equation are the -terms, and the third terms are for the variable .
The above system has three equations in three variables and therefore the augmented matrix has dimension , hence taking the form
We will begin populating the entries of this matrix on a column-by-column basis, beginning with the left-most column. The first term of each equation in (3) is the -term. For the first equation the coefficient is 1, for the second equation the coefficient is 0, and for the third equation the coefficient is . Thus, we can complete the first column of the augmented matrix as follows:
Now we examine the -terms in each equation of (3). For the first, second, and third equations, the coefficients of these terms are all equal to 1. Therefore, the second column of the augmented matrix can be completed, giving
To now complete the third column of this matrix, we examine the -terms in equation (3). The coefficients of all of these terms are equal to , which allows us to speedily complete the third column of the augmented matrix as follows:
For the final column, we examine the terms which appear to the right of the equals signs in the system of linear equations in (3). These are 5, 2, and 2 respectively. Therefore, the remaining blank entries of the augmented matrix are populated with these values, giving the final result of
The previous questions were posed with the variables already being written in the same order for all equations of the linear system. There is no particular reason for this to be the case, and it is possible/likely that at some point we will have to work with a system of linear equations that is not written in a particularly helpful way. If this is the case, then there is essentially only one additional step required as part of the method for finding the augmented matrix. This step is simple and is assisted by consistently coloring each variable, as with the previous questions.
Example 4: Finding the Augmented Matrix for a System of Three Linear Equations in Three Variables Where Rearrangement Is Required
Find the augmented matrix for the following system of equations:
By examining the system of linear equations given above, we can see that this is not written in the most helpful format. In order to minimize our chance of making a mistake when creating the augmented matrix, we first rewrite this system of linear equations by coloring the -variables in pink, the -variables in blue, and the -variables in green. This gives
As we can see by looking at the colors, this system of equation is not aligned in a way that will assist us in creating the augmented matrix. Therefore, we choose to rewrite this system of linear equations in a format that is more helpful. This step will not alter any properties of the system of linear equations, which will be identical to that given above. With horizontal alignment imposed on the basis of color, the equivalent system is
We can now complete the augmented matrix with recourse to this better-formatted expression. The system has three equations in three variables and hence the augmented matrix is of dimension
We begin by populating the first column of this matrix. To do this, we examine the coefficients of the -terms in (4). For the first, second, and third equations, these coefficients are 8, 3, and 0 respectively. Therefore the augmented matrix takes the form
Continuing, we now look at the -terms that appear in (4). Reading from top to bottom, these are 0, 6, and . The second column of the augmented matrix is completed with these entries, giving
To complete the third column, we look at the -terms that appear in the system of linear equations in (4). For the first, second, and third equation these are , 0, and 7 respectively. The third column of the augmented matrix can now be completed, giving
The final step can be completed by examining the terms that appear on the right-hand side of the equals signs in each equation of (4). These are 7, 0, and , respectively, which gives the final augmented matrix
In the final example of this explainer, we will be presented with an augmented matrix and asked to write this as a corresponding system of linear equations. We gave one example above of this type where there were two equations and two variables in the system of linear equations. In the following example, we will increase the complexity by working with a system of three linear equations in three variables.
Example 5: Finding the System of Linear Equations Represented by 3 × 4 Augmented Matrix
From the augmented matrix find the system of equations.
We begin by assuming that the variables corresponding to the first, second, and third columns should be labeled as , , and respectively. This is an arbitrary choice of variables and means that we should populate the missing entries in the following system:
The coefficients of the -terms in these three equations can be immediately written by examining the left-most column of the augmented matrix. In order, the entries of the first column are 2, 0, and . Therefore, the system of equations becomes where we have suppressed the -term in the second equation due to the coefficient of this term being 0. We now look at the second column of the augmented matrix, the coefficients of which (in order) are 0, 4, and . This means that we now know the coefficients of the -terms in our system of equations, giving where the -term in the first equation has been omitted because the coefficient of this term is 0. Now, we will write the coefficients of the -terms in each equation. The third column of the augmented matrix has entries , , and 0. By entering these into the above system of equations and bringing all terms together, we therefore have
The final set of missing values can be obtained from the right-most column in the augmented matrix. In order, these entries are 5, 5, and 0. This gives the complete set of linear equations as follows:
We have seen in this explainer that it is usually a simple process to switch between a system of linear equations and the corresponding augmented matrix. The one caveat to this approach is that the system of linear equations should be written in a way so that the variables always appear in the same order for each individual equation of this system. If this is the case, then switching between the system of linear equations and the augmented matrix is essentially trivial. Once a system of linear equations is written into the correct augmented matrix, then we can undertake solving the system of equations by manipulating the augmented matrix using row operations as part of Gauss–Jordan elimination.
- Consider a general system of linear equations, as follows: Then, the “coefficient matrix” is and the “augmented matrix” is
- For a system of linear equations with equations in variables, the coefficient matrix has rows and columns and therefore has dimension .
- The augmented matrix of such a system has rows and columns, meaning that it has dimension .
- When writing a system of linear equations as an augmented matrix, it is essential that the variables appear in the same order for each equation before the augmented matrix is populated with the coefficients of this system.
- Coloring variables can help to understand whether the system of linear equations is already written in a helpful form.
- If a quantity appears that is neither a variable nor a coefficient of a variable, then it should appear on the right-hand side of the equals sign before the augmented matrix is created.
- If a variable does not appear in one of the equations, then the coefficient of this variable is 0 and the augmented matrix should have a value of 0 in the corresponding entry.