Determine The Values Of The Parameter S For Which The System Has A Unique Solution, And Describe The

Determine The Values Of The Parameter S For Which The System Has A Unique Solution, And Describe The

Understanding when a system of equations has a unique solution is fundamental in linear algebra and its applications across engineering, physics, economics, and computer science. The parameter \(S\) often appears as a variable or a coefficient within the system, influencing the system's solvability. This comprehensive guide aims to help you analyze such systems, determine the specific values of \(S\) that guarantee a unique solution, and interpret the implications of these findings.

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Introduction to Systems of Linear Equations and Parameters

A system of linear equations consists of multiple equations involving the same set of variables. The general form can be expressed as:

\[
A \mathbf{x} = \mathbf{b}
\]

where:

• \(A\) is the coefficient matrix,


• \(\mathbf{x}\) is the vector of variables,


• \(\mathbf{b}\) is the constant vector.

The inclusion of a parameter \(S\) typically appears within the coefficient matrix \(A\) or the vector \(\mathbf{b}\), transforming the system into a parametric family of systems:

\[
A(S) \mathbf{x} = \mathbf{b}(S)
\]

The goal is to determine the values of \(S\) for which the system admits a unique solution, meaning that for those specific values, the system's solution exists and is uniquely determined.

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Fundamental Concepts for Analyzing System Solutions

Before delving into the parametrized system, it is essential to understand key concepts:

1. Consistency of the System

• A system is consistent if at least one solution exists.
• It is inconsistent if no solutions exist.

2. Number of Solutions

• Unique solution: exactly one solution.
• Infinite solutions: infinitely many solutions, typically occurring when the system is underdetermined or has dependent equations.

3. Role of the Coefficient Matrix Determinant

• The determinant of the coefficient matrix, \(\det(A)\), plays a critical role.
• If \(\det(A) \neq 0\), the system has a unique solution.
• If \(\det(A) = 0\), the system may have either no solutions or infinitely many solutions.

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Analyzing a Parametric System: Step-by-Step Approach

To determine the specific values of \(S\) for which the system has a unique solution, follow these steps:

Step 1: Write the System in Matrix Form

Express the system explicitly, noting the dependence on \(S\). For example:

\[
\begin{cases}
a{11}(S) x1 + a{12}(S) x2 + \dots + a{1n}(S) xn = b_1(S) \\
a{21}(S) x1 + a{22}(S) x2 + \dots + a{2n}(S) xn = b_2(S) \\
\vdots \\
a{m1}(S) x1 + a{m2}(S) x2 + \dots + a{mn}(S) xn = b_m(S)
\end{cases}
\]

Step 2: Identify the Coefficient Matrix \(A(S)\)

Construct the matrix:

\[
A(S) = \begin{bmatrix}
a{11}(S) & a{12}(S) & \dots & a_{1n}(S) \\
a{21}(S) & a{22}(S) & \dots & a_{2n}(S) \\
\vdots & \vdots & \ddots & \vdots \\
a{m1}(S) & a{m2}(S) & \dots & a_{mn}(S)
\end{bmatrix}
\]

and the vector:

\[
\mathbf{b}(S) = \begin{bmatrix}
b_1(S) \\
b_2(S) \\
\vdots \\
b_m(S)
\end{bmatrix}
\]

Step 3: Compute \(\det(A(S))\)

Determine the determinant as a function of \(S\). This often involves algebraic expansion or applying properties of determinants for matrices with parametric entries.

Step 4: Find the Values of \(S\) Making \(\det(A(S)) \neq 0\)

• The system has a unique solution whenever \(\det(A(S)) \neq 0\).
• Solve the inequality \(\det(A(S)) \neq 0\) to identify the permissible values of \(S\).

Step 5: Verify the Consistency of the System at These Values

• Even if \(\det(A(S)) \neq 0\), ensure that the system is consistent for these \(S\) by checking if the augmented matrix \([A(S) | \mathbf{b}(S)]\) has the same rank as \(A(S)\).
• Typically, for \(\det(A(S)) \neq 0\), the system is consistent and has a unique solution.

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Practical Examples and Application

Let's analyze a concrete example to understand the process better.

Example System:

\[ \begin{cases} x + 2y + Sz = 4 \\ 2x + (S+1)y + 3z = 7 \\ 3x + 4y + (2S)z = 10 \end{cases} \]

Objective: Find the values of \(S\) for which this system has a unique solution.

Solution Steps:

1. Construct the coefficient matrix: \[ A(S) = \begin{bmatrix} 1 & 2 & S \\ 2 & S+1 & 3 \\ 3 & 4 & 2S \end{bmatrix} \]
2. Calculate the determinant: \[ \det(A(S)) = \begin{vmatrix} 1 & 2 & S \\ 2 & S+1 & 3 \\ 3 & 4 & 2S \end{vmatrix} \]
3. Compute \(\det(A(S))\): \[ \det(A(S)) = 1 \cdot \begin{vmatrix} S+1 & 3 \\ 4 & 2S \end{vmatrix} - 2 \cdot \begin{vmatrix} 2 & 3 \\ 3 & 2S \end{vmatrix} + S \cdot \begin{vmatrix} 2 & S+1 \\ 3 & 4 \end{vmatrix} \] Calculate minors: \[ \begin{aligned} & \begin{vmatrix} S+1 & 3 \\ 4 & 2S \end{vmatrix} = (S+1)(2S) - 4 \times 3 = 2S(S+1) - 12 = 2S^2 + 2S - 12 \\ & \begin{vmatrix} 2 & 3 \\ 3 & 2S \end{vmatrix} = 2 \times 2S - 3 \times 3 = 4S - 9 \\ & \begin{vmatrix} 2 & S+1 \\ 3 & 4 \end{vmatrix} = 2 \times 4 - (S+1) \times 3 = 8 - 3S - 3 = 5 - 3S \end{aligned} \] Now substitute back: \[ \det(A(S)) = 1 \times (2S^2 + 2S - 12) - 2 \times (4S - 9) + S \times (5 - 3S) \] Simplify: \[ = 2S^2 + 2S - 12 - 8S + 18 + 5S - 3S^2 \] Combine like terms: \[ (2S^2 - 3S^2) + (2S - 8S + 5S) + (-12 + 18) = -S^2 - 1S + 6 \] Therefore: \[ \det(A(S)) = -S^2 - S + 6 \]
   1. Find values of \(S\) such that \(\det(A(S)) \neq 0\): \[ -S^2 - S + 6 \neq 0 \] or equivalently: \[ S^2 + S - 6 \neq 0 \] Factor: \[ (S + 3)(S - 2) \neq 0 \] so: \[ S \neq -3, \quad S \neq 2 \]

   Conclusion:

   • The system has a unique solution for all real values of \(S\) except \(S = -3\) and