Solving System Of Equations With Augmented Matrices

In linear algebra, augmented matrices provide a compact and efficient way to represent and solve systems of linear equations. This article delves into the process of solving a system of equations using augmented matrices, focusing on the interpretation of row-reduced echelon form (RREF) to determine the solution. We will analyze a specific example where an augmented matrix is reduced to RREF, and then deduce the solution, or lack thereof, for the original system of equations. The concepts of consistency, inconsistency, and the implications of a row of zeros in the RREF will be thoroughly explored. We aim to provide a comprehensive understanding of how augmented matrices serve as a powerful tool in solving linear systems, enhancing the reader's grasp of fundamental concepts in linear algebra.

Understanding Augmented Matrices

An augmented matrix is a representation of a system of linear equations, where the coefficients of the variables and the constants are arranged in a matrix format. The coefficients of the variables form the main part of the matrix, and the constants are appended as an additional column, separated by a vertical line (often implied). This representation allows us to perform row operations, which are elementary operations that transform the matrix while preserving the solution set of the system. Row operations include swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. The goal of these operations is to transform the augmented matrix into its row-reduced echelon form (RREF), a form from which the solution to the system can be easily read. Understanding augmented matrices is crucial for efficiently solving linear systems and forms a cornerstone of linear algebra.

Row-Reduced Echelon Form (RREF)

The row-reduced echelon form (RREF) is a specific form of a matrix that simplifies the process of solving systems of linear equations. A matrix is in RREF if it satisfies the following conditions:

1. All rows consisting entirely of zeros are at the bottom of the matrix.
2. The leading coefficient (the first nonzero number) of each nonzero row is 1.
3. The leading 1 in each nonzero row is to the right of the leading 1 in the row above it.
4. Each column containing a leading 1 has zeros in all other entries.

Transforming an augmented matrix into RREF involves applying a sequence of row operations, which systematically eliminate variables and simplify the equations. The RREF provides a clear and concise representation of the solution set of the system. For example, if the RREF has a row of the form [0 0 ... 0 | 1], it indicates that the system is inconsistent and has no solution. Conversely, if the RREF has a unique solution, it can be directly read from the matrix. The RREF is a fundamental concept in linear algebra, allowing for efficient and accurate solutions to systems of linear equations.

Analyzing the Given Augmented Matrix

We are given an augmented matrix that has been row-reduced to the following form:

$$\left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 1 \end{array}\right]$$

This matrix represents a system of linear equations in three variables. The first three columns correspond to the coefficients of the variables, and the last column represents the constants. To understand the implications of this RREF, we need to interpret each row as an equation. The first row, [1 0 0 | -2], corresponds to the equation 1x + 0y + 0z = -2, which simplifies to x = -2. The second row, [0 1 0 | 4], corresponds to the equation 0x + 1y + 0z = 4, which simplifies to y = 4. The third row, [0 0 0 | 1], corresponds to the equation 0x + 0y + 0z = 1, which simplifies to 0 = 1. This last equation is a contradiction, indicating that the system of equations is inconsistent and has no solution. The presence of this contradictory row in the RREF is a clear signal that the original system of equations does not have a solution. In summary, analyzing the given augmented matrix allows us to determine the nature of the solution set of the corresponding system of linear equations.

Determining the Solution

Based on the row-reduced echelon form (RREF) of the augmented matrix,

$$\left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 1 \end{array}\right]$$

we can determine the solution to the original system of equations. The first row indicates that x = -2, and the second row indicates that y = 4. However, the third row, [0 0 0 | 1], represents the equation 0 = 1, which is a contradiction. This contradiction implies that the system of equations is inconsistent, meaning there is no solution that satisfies all three equations simultaneously. In other words, there are no values for the variables that can make all equations in the original system true. Therefore, the solution to the original system of equations is that there is no solution. This outcome is a direct consequence of the contradictory row in the RREF, highlighting the power of augmented matrices in determining the solvability of linear systems.

Implications of a Contradictory Row

A contradictory row, such as [0 0 ... 0 | 1] in the RREF of an augmented matrix, has significant implications for the solution of the corresponding system of linear equations. This row represents an equation of the form 0 = 1, which is mathematically impossible. The presence of such a row indicates that the system is inconsistent, meaning there is no set of values for the variables that can satisfy all equations in the system simultaneously. Geometrically, this could mean that the lines or planes represented by the equations do not intersect at any common point. In practical terms, a contradictory row suggests that there might be an error in the formulation of the system or that the system is modeling a situation that is inherently impossible. Understanding the implications of a contradictory row is crucial for correctly interpreting the RREF of an augmented matrix and determining the nature of the solution set of the linear system.

Conclusion

In conclusion, augmented matrices provide a robust method for solving systems of linear equations. By transforming the augmented matrix into its row-reduced echelon form (RREF), we can easily determine the solution, or lack thereof, for the original system. In the given example, the RREF contained a contradictory row, indicating that the system of equations has no solution. This outcome highlights the importance of understanding the implications of different forms in the RREF, such as rows of zeros or contradictory rows, in determining the nature of the solution set. Augmented matrices are a fundamental tool in linear algebra, enabling efficient and accurate solutions to a wide range of problems involving linear systems. The process of row reduction and interpretation of the RREF are essential skills for anyone working with linear equations and matrices.