Unique Solution Analysis Of Linear Equations System Regardless Of Λ

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In linear algebra, determining whether a system of linear equations has a unique solution is a fundamental problem with wide-ranging applications in various fields, including engineering, physics, economics, and computer science. A system of linear equations can have one of three possible outcomes: a unique solution, infinitely many solutions, or no solution. The nature of the solution set depends on the coefficients of the equations and the constants on the right-hand side. This article delves into the intricacies of determining the conditions for a unique solution in a system of linear equations, focusing on the role of parameters and the use of augmented matrices.

A system of linear equations consists of two or more linear equations involving the same variables. Each linear equation represents a straight line in a two-dimensional space or a hyperplane in higher dimensions. The solution to the system is the set of values for the variables that satisfy all equations simultaneously. Geometrically, this corresponds to the point(s) where the lines or hyperplanes intersect. The existence and uniqueness of solutions are crucial concepts in linear algebra, underpinning many analytical and computational techniques.

The augmented matrix is a powerful tool for analyzing systems of linear equations. It is formed by combining the coefficient matrix (containing the coefficients of the variables) and the constant matrix (containing the constants on the right-hand side) into a single matrix. By performing row operations on the augmented matrix, we can transform the system into a simpler form, such as row-echelon form or reduced row-echelon form, which makes it easier to determine the nature of the solutions.

The Augmented Matrix and Its Significance

The augmented matrix provides a concise and organized representation of a system of linear equations. Consider the system:

2x + 2y = λ
-x + y = s

The augmented matrix for this system is:

[ 2  2 | λ ]
[ -1 1 | s ]

The first two columns represent the coefficients of the variables x and y, respectively, while the last column represents the constants on the right-hand side. The vertical line separates the coefficient matrix from the constant matrix. Row operations, which are elementary operations that do not change the solution set of the system, can be applied to the augmented matrix to simplify it. These operations include:

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

By applying these operations strategically, we can transform the augmented matrix into a form that reveals the solution set of the system. The goal is often to achieve row-echelon form or reduced row-echelon form. The row-echelon form has the following properties:

  • All nonzero rows (rows with at least one nonzero element) are above any rows of all zeros.
  • 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.
  • All entries in a column below a leading entry are zeros.

The reduced row-echelon form is a more stringent form with the additional properties:

  • The leading entry in each nonzero row is 1.
  • Each leading 1 is the only nonzero entry in its column.

Conditions for a Unique Solution

A system of linear equations has a unique solution if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix and is equal to the number of variables. The rank of a matrix is the maximum number of linearly independent rows (or columns) in the matrix. Linear independence means that no row (or column) can be written as a linear combination of the other rows (or columns).

For the given system, the coefficient matrix is:

[ 2  2 ]
[ -1 1 ]

and the augmented matrix is:

[ 2  2 | λ ]
[ -1 1 | s ]

To determine whether the system has a unique solution regardless of the value of λ, we need to analyze the ranks of these matrices. If the rank of the coefficient matrix is equal to the rank of the augmented matrix and is equal to the number of variables (which is 2 in this case), then a unique solution exists. If the ranks are not equal, or if the rank is less than the number of variables, then the system either has infinitely many solutions or no solution.

Analyzing the Rank of the Coefficient Matrix

The coefficient matrix is:

[ 2  2 ]
[ -1 1 ]

To find the rank, we can perform row operations to reduce the matrix to row-echelon form. We can start by adding 1/2 times the first row to the second row:

[ 2  2 ]
[ 0  2 ]

Now, we have two nonzero rows, and the leading coefficients are in different columns. This means that the rows are linearly independent, and the rank of the coefficient matrix is 2.

Analyzing the Rank of the Augmented Matrix

The augmented matrix is:

[ 2  2 | λ ]
[ -1 1 | s ]

We apply the same row operations to the augmented matrix as we did to the coefficient matrix. Adding 1/2 times the first row to the second row gives:

[ 2  2 | λ ]
[ 0  2 | s + λ/2 ]

Now, the augmented matrix has two nonzero rows. The rank of the augmented matrix will be 2 if the second row is not a zero row. The second row is zero if and only if:

s + λ/2 = 0

which implies:

s = -λ/2

If s ≠ -λ/2, the rank of the augmented matrix is 2. If s = -λ/2, the rank of the augmented matrix could be less than 2, depending on whether the first row becomes a multiple of the second row.

Determining the Conditions for a Unique Solution Regardless of λ

For a unique solution to exist, the rank of the coefficient matrix must equal the rank of the augmented matrix, and both must equal the number of variables. We have already established that the rank of the coefficient matrix is 2. Therefore, the augmented matrix must also have a rank of 2 for a unique solution to exist. This requires that the second row of the row-echelon form of the augmented matrix is nonzero, which means:

s + λ/2 ≠ 0

However, the question asks whether the system can have a unique solution regardless of the value of λ. This means that even if we vary λ, we should still have a unique solution. The condition s ≠ -λ/2 depends on λ, so it does not guarantee a unique solution for all λ.

Let's revisit the row-echelon form of the augmented matrix:

[ 2  2 | λ ]
[ 0  2 | s + λ/2 ]

For the rank of the augmented matrix to be 2 regardless of λ, the second row must always be nonzero. This can only happen if the coefficient in the second row and second column is nonzero, which it is (it's 2). However, the term s + λ/2 can be zero for some value of λ (specifically, when λ = -2s). Therefore, the system cannot have a unique solution regardless of the value of λ.

In summary, the system of linear equations given by the augmented matrix:

[ 2  2 | λ ]
[ -1 1 | s ]

does not have a unique solution regardless of the value of λ. The condition for a unique solution, s ≠ -λ/2, depends on the specific value of λ, meaning there will always be a value of λ for which the system does not have a unique solution (either infinitely many solutions or no solution). This analysis highlights the importance of understanding the rank of matrices and the conditions under which systems of linear equations possess unique solutions. The augmented matrix is a powerful tool for analyzing the solution space of linear systems, and row operations allow us to systematically determine the nature of the solutions.

In conclusion, while the system can have a unique solution for certain values of λ, it does not hold true for all possible values, thereby answering the initial question with a definitive no.