Consistent System Of Equations Augmented Matrix Method

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#h1 Introduction In the realm of linear algebra, solving systems of equations is a fundamental task. Augmented matrices provide a powerful and systematic way to tackle these problems. This article delves into the process of determining whether a system of equations represented by an augmented matrix is consistent, meaning it has at least one solution. We will explore the underlying concepts, demonstrate the steps involved, and provide a comprehensive understanding of this essential technique. In this article, we will consider the system of equations represented by the augmented matrix:

[0 1 0 1 | 1]
[2 5 -4 1 | 1]

We aim to determine whether this system is consistent. Let's embark on this journey by first understanding the basics of augmented matrices and their connection to systems of equations.

#h2 Understanding Augmented Matrices

Before we dive into the solution, let's establish a solid understanding of augmented matrices. An augmented matrix is a compact way to represent a system of linear equations. It combines the coefficients of the variables and the constants into a single matrix. Each row in the matrix corresponds to an equation in the system, and each column (except the last) corresponds to a variable. The last column represents the constants on the right-hand side of the equations.

For instance, the given augmented matrix:

[0 1 0 1 | 1]
[2 5 -4 1 | 1]

represents the following system of linear equations:

0x₁ + 1x₂ + 0x₃ + 1x₄ = 1
2x₁ + 5x₂ - 4x₃ + 1x₄ = 1

This can be simplified to:

x₂ + x₄ = 1
2x₁ + 5x₂ - 4x₃ + x₄ = 1

Now, to determine if the system is consistent, we need to find out if there is at least one solution that satisfies both equations simultaneously. We will achieve this by transforming the augmented matrix into its row-echelon form using elementary row operations. The row-echelon form will reveal the nature of the solutions, if any, and allow us to conclude whether the system is consistent or inconsistent. Let's move on to the next step: transforming the matrix into row-echelon form.

#h3 Transforming to Row-Echelon Form

The key to determining the consistency of the system lies in transforming the augmented matrix into its row-echelon form. This form has specific characteristics that make it easy to analyze the solutions. A matrix is in row-echelon form if:

  1. All nonzero rows (rows with at least one nonzero element) are above any rows of all zeros.
  2. The leading coefficient (the first nonzero number from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it.
  3. All entries in a column below a leading entry are zeros.

To achieve this form, we employ elementary row operations. These operations do not change the solution set of the system and include:

  1. Swapping two rows.
  2. Multiplying a row by a nonzero constant.
  3. Adding a multiple of one row to another row.

Let's apply these operations to our given matrix:

[0 1 0 1 | 1]
[2 5 -4 1 | 1]

First, we swap Row 1 and Row 2 to get a nonzero entry in the top-left position:

[2 5 -4 1 | 1]
[0 1 0 1 | 1]

Now, the matrix is already in row-echelon form because:

  1. Both rows are nonzero.
  2. The leading coefficient in the first row (2) is to the left of the leading coefficient in the second row (1).
  3. All entries below the leading coefficients are zero.

With the matrix in row-echelon form, we can analyze the system for consistency. The next section will delve into this analysis.

#h4 Analyzing for Consistency

Now that we have the augmented matrix in row-echelon form:

[2 5 -4 1 | 1]
[0 1 0 1 | 1]

We can analyze it to determine if the system of equations is consistent. A system is consistent if it has at least one solution. In the context of row-echelon form, inconsistency arises when we encounter a row that represents a contradictory equation, such as 0 = c, where c is a nonzero constant. This implies that there is no solution that satisfies the system.

Looking at our row-echelon form, we can rewrite the matrix back into its equation form:

2x₁ + 5x₂ - 4x₃ + x₄ = 1
x₂ + x₄ = 1

Notice that there is no row in the matrix that leads to a contradiction. Both equations are valid and do not result in a statement like 0 = c (where c ≠ 0). This indicates that the system is consistent.

Furthermore, we can observe that we have two equations and four variables (x₁, x₂, x₃, x₄). This means we have more variables than equations, which generally suggests that there are infinitely many solutions or no solution. However, since we didn't encounter a contradiction, we can conclude that there are infinitely many solutions in this case.

Therefore, based on our analysis of the row-echelon form, the system of equations is consistent. In the following section, we will provide a conclusive answer and summarize our findings.

#h5 Conclusion

In conclusion, by transforming the given augmented matrix:

[0 1 0 1 | 1]
[2 5 -4 1 | 1]

into its row-echelon form:

[2 5 -4 1 | 1]
[0 1 0 1 | 1]

we have successfully determined that the system of equations it represents is consistent. This means that there is at least one solution that satisfies all equations in the system.

Our analysis revealed that no row in the row-echelon form leads to a contradictory equation, such as 0 = c (where c is a nonzero constant). Furthermore, the presence of more variables than equations suggests that the system has infinitely many solutions.

This step-by-step process of transforming an augmented matrix to row-echelon form and analyzing the resulting matrix is a fundamental technique in linear algebra for solving systems of equations and determining their consistency. By mastering this method, you can confidently tackle a wide range of linear algebra problems.

#h2 Further Exploration

To deepen your understanding of solving systems of equations, here are some avenues for further exploration:

  • Reduced Row-Echelon Form: While row-echelon form is useful for determining consistency, the reduced row-echelon form provides even more information. In this form, the leading coefficient in each nonzero row is 1, and all other entries in the column containing a leading coefficient are 0. Transforming the matrix to reduced row-echelon form allows you to directly read off the solutions.
  • Gaussian Elimination: This is the algorithm used to transform a matrix into row-echelon form. Understanding the steps involved in Gaussian elimination will solidify your grasp of the process.
  • Gauss-Jordan Elimination: This algorithm extends Gaussian elimination to produce the reduced row-echelon form.
  • Applications of Systems of Equations: Systems of equations have countless applications in various fields, including engineering, physics, economics, and computer science. Explore how these systems are used to model real-world phenomena.

By delving into these topics, you will gain a more comprehensive understanding of systems of equations and their significance in mathematics and beyond.