For The Following System To Be Consistent

For a system to be consistent, it must possess at least one solution; in simpler terms, a set of values for the variables that satisfy all equations within the system simultaneously. This concept, pivotal in linear algebra and applicable across various scientific and engineering disciplines, dictates the solvability of systems of equations. The consistency of a system hinges on the relationships between the equations and the existence of solutions that make all equations true at once.

Understanding System Consistency

A system of equations is consistent if there is at least one set of values that, when substituted for the unknowns (variables), satisfies every equation in the system. Conversely, an inconsistent system has no such solution. Determining whether a system is consistent is crucial before attempting to solve it.

Consider these foundational aspects:

• Linear Equations: These equations, when graphed, produce straight lines. In two variables, the intersection point (if it exists) represents the solution.
• Number of Equations and Variables: A system can have fewer equations than variables (underdetermined), more equations than variables (overdetermined), or an equal number. Each scenario affects the likelihood of a consistent solution.
• Geometric Interpretation: In two dimensions, consistency implies that lines intersect, are the same line, or, in higher dimensions, planes and hyperplanes have at least one common point.

Methods to Determine Consistency

Several methods can ascertain whether a system of equations is consistent. Here are some prominent techniques:

1. Gaussian Elimination and Row Echelon Form

Gaussian elimination is a systematic approach to transform a system of linear equations into row echelon form using elementary row operations. These operations include:

• Swapping two rows.
• Multiplying a row by a non-zero constant.
• Adding a multiple of one row to another.

Once in row echelon form, the consistency can be determined by examining the last row. If the last row has all zeros except for the last column (the constant term), and that entry is non-zero, the system is inconsistent.

Example:

Consider the system:

x + y = 3
2x + 2y = 5

Transforming this into an augmented matrix:

[ 1  1 | 3 ]
[ 2  2 | 5 ]

Applying row operation R2 -> R2 - 2R1:

[ 1  1 | 3 ]
[ 0  0 | -1 ]

The second row implies 0 = -1, which is impossible, hence the system is inconsistent.

2. Rank of the Matrix

The rank of a matrix is the number of linearly independent rows or columns. For a system of equations represented as Ax = b, where A is the coefficient matrix, x is the variable vector, and b is the constant vector:

• The system is consistent if and only if the rank of A equals the rank of the augmented matrix [A|b].
• If rank(A) < rank([A|b]), the system is inconsistent.

Example:

Consider the system:

x + y = 3
2x + 2y = 6

Matrix A is:

[ 1  1 ]
[ 2  2 ]

Augmented matrix [A|b] is:

[ 1  1 | 3 ]
[ 2  2 | 6 ]

Rank(A) = 1 and Rank([A|b]) = 1. Since both ranks are equal, the system is consistent.

3. Determinant of the Matrix

For a square matrix (equal number of equations and variables), the determinant provides valuable information.

• If det(A) ≠ 0, the system Ax = b has a unique solution and is consistent.
• If det(A) = 0, the system may be consistent with infinitely many solutions or inconsistent. Further analysis is needed, often involving the rank of the matrix.

Example:

Consider the system:

x + y = 3
x - y = 1

Matrix A is:

[ 1  1 ]
[ 1 -1 ]

det(A) = (1*-1) - (1*1) = -2. Since det(A) ≠ 0, the system has a unique solution and is consistent.

4. Rouché–Capelli Theorem

The Rouché–Capelli theorem (also known as the Rouché–Frobenius theorem) provides a comprehensive criterion for determining the solvability of a system of linear equations. It states:

A system of linear equations with n variables is consistent if and only if the rank of the coefficient matrix A is equal to the rank of the augmented matrix [A|b]. Moreover:

• If rank(A) = rank([A|b]) = n, the system has a unique solution.
• If rank(A) = rank([A|b]) < n, the system has infinitely many solutions.
• If rank(A) < rank([A|b]), the system is inconsistent.

This theorem elegantly encapsulates all consistency scenarios.

Conditions for Consistency

To further refine the criteria for system consistency, consider the following specific conditions:

1. Underdetermined Systems

An underdetermined system has fewer equations than variables. Such systems can be consistent and typically have infinitely many solutions, provided the equations are linearly independent. However, if there's linear dependence that leads to contradictions, the system can be inconsistent.

Example:

x + y + z = 5

Here, there's one equation and three variables. It’s an underdetermined system. If we introduce another equation:

x + y + z = 5
2x + 2y + 2z = 10

The system is consistent with infinite solutions as the second equation is just a multiple of the first.

But, if the second equation is:

x + y + z = 5
2x + 2y + 2z = 11

The system becomes inconsistent.

2. Overdetermined Systems

An overdetermined system has more equations than variables. These systems are often inconsistent because each additional equation imposes a constraint that the variables might not be able to satisfy simultaneously. However, there are cases where overdetermined systems can be consistent, particularly if some equations are linear combinations of others.

Example:

x + y = 3
x - y = 1
2x = 4

Solving the first two equations gives x = 2 and y = 1. The third equation, 2x = 4, is satisfied by x = 2, thus the system is consistent.

However, if the third equation were 2x = 5, the system would be inconsistent.

3. Homogeneous Systems

A homogeneous system is one where all constant terms are zero (i.e., Ax = 0). Homogeneous systems are always consistent because they always have the trivial solution x = 0. The key question is whether they have non-trivial solutions.

• If det(A) ≠ 0, the only solution is the trivial solution.
• If det(A) = 0, there are infinitely many solutions, including non-trivial ones.

Example:

x + y = 0
x - y = 0

The trivial solution is x = 0 and y = 0. The determinant of the coefficient matrix is -2, so this is the only solution.

4. Linear Independence

Linear independence of equations is crucial for ensuring the consistency of a system. If one or more equations can be derived from others through linear combinations, they do not add new constraints, and the system's consistency remains unaffected. Conversely, if an equation introduces a new, independent constraint that contradicts the existing equations, the system becomes inconsistent.

Example:

Consider the equations:

x + y = 3
2x + 2y = 6

The second equation is just twice the first equation, thus they are linearly dependent and the system is consistent with infinitely many solutions.

Practical Applications

Understanding the consistency of systems of equations is fundamental in various fields:

1. Engineering

In structural analysis, systems of equations are used to model the forces and stresses within structures. Ensuring that these systems are consistent is critical for designing safe and stable structures.

2. Economics

In econometrics, systems of equations represent economic models. These models must be consistent to yield meaningful and reliable predictions about economic behavior.

3. Computer Graphics

In computer graphics, systems of linear equations are used to perform transformations, projections, and rendering operations. Ensuring consistency is crucial for generating accurate and visually coherent images.

4. Data Analysis

In data analysis, systems of equations can arise in regression analysis or when fitting models to data. Determining the consistency of these systems helps in validating the models and ensuring the reliability of the results.

5. Network Analysis

In network analysis, systems of equations describe flow conservation in networks, such as electrical circuits or traffic networks. The consistency of these systems ensures that the network behaves as expected.

Examples of Determining Consistency

Example 1: Using Gaussian Elimination

Consider the system:

x + y + z = 6
x - y + z = 2
2x + y - z = 1

Augmented matrix:

[ 1  1  1 | 6 ]
[ 1 -1  1 | 2 ]
[ 2  1 -1 | 1 ]

Applying Gaussian elimination:

R2 -> R2 - R1:

[ 1  1  1 | 6 ]
[ 0 -2  0 | -4 ]
[ 2  1 -1 | 1 ]

R3 -> R3 - 2R1:

[ 1  1  1 | 6 ]
[ 0 -2  0 | -4 ]
[ 0 -1 -3 | -11 ]

R3 -> R3 - (1/2)R2:

[ 1  1  1 | 6 ]
[ 0 -2  0 | -4 ]
[ 0  0 -3 | -9 ]

From the row echelon form, we can solve:

-3z = -9 => z = 3
-2y = -4 => y = 2
x + 2 + 3 = 6 => x = 1

The system is consistent with a unique solution: x = 1, y = 2, z = 3.

Example 2: Using Rank

Consider the system:

x + y = 5
2x + 2y = 10

Matrix A:

[ 1  1 ]
[ 2  2 ]

Augmented matrix [A|b]:

[ 1  1 | 5 ]
[ 2  2 | 10 ]

Rank(A) = 1 Rank([A|b]) = 1

Since Rank(A) = Rank([A|b]) < number of variables (2), the system is consistent with infinitely many solutions.

Example 3: An Inconsistent System

Consider the system:

x + y = 5
2x + 2y = 12

Matrix A:

[ 1  1 ]
[ 2  2 ]

Augmented matrix [A|b]:

[ 1  1 | 5 ]
[ 2  2 | 12 ]

Rank(A) = 1 Rank([A|b]) = 2

Since Rank(A) < Rank([A|b]), the system is inconsistent.

Advanced Topics

1. Least Squares Solutions

In many practical applications, especially with overdetermined systems, a perfect solution does not exist. In such cases, the concept of a least squares solution becomes relevant. This approach seeks a solution that minimizes the sum of the squares of the residuals (the differences between the observed and predicted values).

2. Condition Number

The condition number of a matrix provides a measure of how sensitive the solution of a system of equations is to changes in the input data. A high condition number indicates that the system is ill-conditioned, meaning small changes in the coefficients can lead to large changes in the solution.

3. Singular Value Decomposition (SVD)

Singular Value Decomposition (SVD) is a powerful technique used to analyze and solve systems of linear equations, especially those that are singular or ill-conditioned. SVD decomposes a matrix into three matrices, allowing for the determination of the rank, null space, and range of the matrix, which are all crucial for understanding the consistency and solvability of the system.

Conclusion

Determining whether a system of equations is consistent is a fundamental step in solving mathematical problems across various disciplines. Methods such as Gaussian elimination, rank determination, and the Rouché–Capelli theorem provide robust techniques to assess consistency. Understanding the underlying principles and conditions for consistency ensures that solutions, if they exist, are valid and meaningful.