Augmented Matrices

You can begin studying specific types of matrices now that you've learned a little bit about matrices (including how a matrix is any rectangular array of numbers enclosed by brackets). For example, there are row matrices, column matrices, square matrices, zero matrices, and more, depending on the contents and dimensions of each matrix. Each of these offers specific use cases and properties that you'll need to be aware of.

One of the most practical applications of matrices is how they can be used to solve systems of linear equations, making matrices related to systems of linear equations extremely important. We'll show you two different kinds of matrices that are used to solve systems of linear equations. Let's get started!

What are augmented matrices?

An augmented matrix may be defined as any matrix derived from a system of linear equations. Let's look at an example of the following system of linear equations:

The first step is making sure that all of the equations are written in standard form, or $Ax+By+Cz=D$, where D is a constant term and the rest have variables x, y, and z in that order. This example is already in standard form, so we can move on to the next step.

Next, list the coefficients associated with each variable vertically such that each of the first three columns aligns with x, y, and z in sequence. If an equation doesn't have one or more variables (like the third one above lacking z), sub in a zero. The fourth column should be aligned to the constants in the equation above and may optionally be separated with a dotted line to indicate that no variables are present. That gives us the following:

Put the numbers in brackets and you have successfully created an augmented matrix! That wasn't so bad, was it?

How are augmented matrices related to coefficient matrices?

Like augmented matrices, coefficient matrices are also related to systems of linear equations. However, the constant terms are excluded from a coefficient matrix, leaving only the columns with variables. Otherwise, they are created in the exact same way. Let's look at an example:

All three equations are written in the standard form already, so all we have to do is create our columns. Here's what that looks like:

The third equation doesn't have an x variable, so don't forget to sub in the zero!

Basically, you'll want to line up the coefficients in an x,y,z sequence before creating an augmented matrix or a coefficient matrix, subbing in zeros rather than leaving any variables blank. If you're creating an augmented matrix, the constant terms should appear on the far right, optionally separated with a dotted line. If you're creating a coefficient matrix, the constant terms shouldn't appear at all even though they appear in the system of linear equations you're working with. Either way, the numbers should be enclosed in brackets to indicate that they are a matrix.

Why are augmented matrices important?

As previously noted, you can solve systems of linear equations using matrices. To do so, you need to understand matrix row operations and the identity matrix, among other concepts. This will probably be one of the first things you do with them.

You can also use matrices to do basic algebraic operations like adding and subtracting, multiplying (if the sizes of the matrices are the same), and even dividing with inverse matrices. These operations will expand your quantitative problem-solving skills substantially.

Advanced mathematicians also use matrices to perform linear transformations, but you probably don't have to worry about that yet. Still, it's worth keeping in mind as an application you might see down the line.

Augmented matrices practice questions

a. Create an augmented matrix for the following system of linear equations:

The augmented matrix is:

b. Create an augmented matrix for the following system of linear equations:

The augmented matrix is:

c. Create a coefficient matrix for the following system of linear equations:

The coefficient matrix is:

d. Create a coefficient matrix for the following system of linear equations:

The coefficient matrix is:

(Note that the equations were NOT written in the standard form)

Topics related to the Augmented Matrices

Matrix Row Operations

Solving Systems of Linear Equations Using Matrices

Inverse Matrices

Flashcards covering the Augmented Matrices

Algebra II Flashcards

College Algebra Flashcards

Practice tests covering the Augmented Matrices

Algebra II Diagnostic Tests

College Algebra Diagnostic Tests

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It's rare for students at this level of mathematics to be introduced to a foundational concept that later coursework will build on, but that's exactly what matrices are. If you or your student don't understand how to work with them, you may feel lost when your instructor gets into their practical applications. Fortunately, you don't need to tackle this concept alone. A 1-on-1 tutor can patiently explain how augmented matrices work until it finally clicks for you. Please reach out to a friendly Educational Director at Varsity Tutors to learn more about the benefits of tutoring or to sign up today.