Example of an Augmented Matrix

Oct 13, 2025 by windailys.com 9 views

Introduction

In linear algebra, an augmented matrix is a matrix that represents a system of linear equations. It is a powerful tool for solving systems of equations and is widely used in various fields such as mathematics, physics, and engineering. In this article, we will discuss how to rewrite an augmented matrix to represent a system of equations.

What is an Augmented Matrix?

An augmented matrix is a matrix that combines the coefficients of the variables in a system of linear equations with the constant terms. It is denoted by the symbol $\left[\begin{array}{ccc|c}\end{array}\right]$ and is used to represent the system of equations in a compact form.

Example of an Augmented Matrix

Consider the following system of linear equations:

\begin{align*} x + y + z &= -1 \\ x + 3y - z &= -9 \\ 3x + 2y &= -2 \end{align*}

The augmented matrix for this system of equations is:

\left[\begin{array}{ccc|c} 1 & 0 & 1 & -1 \\ 1 & 3 & -1 & -9 \\ 3 & 2 & 0 & -2 \end{array}\right]

Rewriting the Augmented Matrix

To rewrite the augmented matrix, we need to perform a series of row operations. In this case, we will perform the operation $-3 R_2 \leftrightarrow R_3$ .

Step 1: Perform the Row Operation

To perform the row operation $-3 R_2 \leftrightarrow R_3$ , we need to multiply the second row by $-3$ and then swap it with the third row.

\left[\begin{array}{ccc|c} 1 & 0 & 1 & -1 \\ 1 & 3 & -1 & -9 \\ 3 & 2 & 0 & -2 \end{array}\right] \xrightarrow{-3 R_2 \leftrightarrow R_3} \left[\begin{array}{ccc|c} 1 & 0 & 1 & -1 \\ 3 & 2 & 0 & -2 \\ 1 & 3 & -1 & -9 \end{array}\right]

Step 2: Simplify the Matrix

After performing the row operation, we need to simplify the matrix by eliminating any zeros that may have been introduced.

\left[\begin{array}{ccc|c} 1 & 0 & 1 & -1 \\ 3 & 2 & 0 & -2 \\ 1 & 3 & -1 & -9 \end{array}\right] \xrightarrow{\text{Simplify}} \left[\begin{array}{ccc|c} 1 & 0 & 1 & -1 \\ 0 & 2 & -3 & 1 \\ 0 & 3 & -2 & -8 \end{array}\right]

Conclusion

In this article, we discussed how to rewrite an augmented matrix to represent a system of equations. We performed a series of row operations to simplify the matrix and eliminate any zeros that may have been introduced. The resulting matrix represents the system of equations in a compact form and can be used to solve for the variables.

Final Answer

The final answer is:

\left[\begin{array}{ccc|c} 1 & 0 & 1 & -1 \\ 0 & 2 & -3 & 1 \\ 0 & 3 & -2 & -8 \end{array}\right]

Discussion

The augmented matrix is a powerful tool for solving systems of linear equations. It allows us to represent the system of equations in a compact form and perform row operations to simplify the matrix. The resulting matrix can be used to solve for the variables and is widely used in various fields such as mathematics, physics, and engineering.

References

Linear Algebra and Its Applications: This book provides a comprehensive introduction to linear algebra and its applications.
Systems of Linear Equations: This article provides a detailed explanation of systems of linear equations and how to solve them using the augmented matrix.
Row Operations: This article provides a detailed explanation of row operations and how to perform them on a matrix.
Augmented Matrix Q&A =====================

Introduction

In our previous article, we discussed how to rewrite an augmented matrix to represent a system of equations. In this article, we will answer some frequently asked questions about augmented matrices and provide additional information to help you understand this topic better.

Q: What is an augmented matrix?

A: An augmented matrix is a matrix that combines the coefficients of the variables in a system of linear equations with the constant terms. It is denoted by the symbol $\left[\begin{array}{ccc|c}\end{array}\right]$ and is used to represent the system of equations in a compact form.

Q: How do I create an augmented matrix?

A: To create an augmented matrix, you need to combine the coefficients of the variables in a system of linear equations with the constant terms. The coefficients of the variables are placed in the matrix, and the constant terms are placed in the last column.

Q: What are some common row operations that can be performed on an augmented matrix?

A: Some common row operations that can be performed on an augmented matrix include:

Multiplying a row by a scalar: This involves multiplying each element in a row by a scalar value.
Adding a multiple of one row to another row: This involves adding a multiple of one row to another row.
Swapping two rows: This involves swapping two rows in the matrix.

Q: How do I perform row operations on an augmented matrix?

A: To perform row operations on an augmented matrix, you need to follow these steps:

Identify the row operation: Determine which row operation you want to perform.
Perform the row operation: Perform the row operation on the augmented matrix.
Simplify the matrix: Simplify the matrix by eliminating any zeros that may have been introduced.

Q: What is the purpose of an augmented matrix?

A: The purpose of an augmented matrix is to represent a system of linear equations in a compact form and to perform row operations to simplify the matrix. This allows us to solve for the variables in the system of equations.

Q: How do I use an augmented matrix to solve a system of linear equations?

A: To use an augmented matrix to solve a system of linear equations, you need to follow these steps:

Create the augmented matrix: Create the augmented matrix by combining the coefficients of the variables in the system of linear equations with the constant terms.
Perform row operations: Perform row operations on the augmented matrix to simplify it and eliminate any zeros that may have been introduced.
Solve for the variables: Solve for the variables in the system of linear equations using the simplified augmented matrix.

Q: What are some common mistakes to avoid when working with augmented matrices?

A: Some common mistakes to avoid when working with augmented matrices include:

Not following the order of operations: Make sure to follow the order of operations when performing row operations.
Not simplifying the matrix: Make sure to simplify the matrix by eliminating any zeros that may have been introduced.
Not solving for the variables: Make sure to solve for the variables in the system of linear equations using the simplified augmented matrix.