Question

Consider the system of linear equations
x + y + z = 5,
x + 2y + λ²z = 9,
x + 3y + λz = μ,
where λ, μ ∈ ℝ.Then, which of the following statement is NOT correct?

• A. System has infinite number of solutions if λ = 1 and μ = 13
• B. System is inconsistent if λ = 1 and μ ≠ 13
• C. System is consistent if λ ≠ 1 and μ = 13
• D. System has a unique solution if λ ≠ 1 and μ ≠ 13

Answer 1

The Correct Option is D

To resolve this issue, we must analyze the provided system of linear equations to ascertain when it possesses a single solution, an infinite number of solutions, or no solution (is inconsistent). The system is defined as follows:

• \( x + y + z = 5 \)
• \( x + 2y + \lambda^2z = 9 \)
• \( x + 3y + \lambda z = \mu \)

We will evaluate each condition presented in the options by examining the distinct scenarios that arise from the values of \(\lambda\) and \(\mu\).

Step 1: System Analysis.

The system can be formulated in matrix form as \( A \mathbf{x} = \mathbf{b} \), where:

\(A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & \lambda^2 \\ 1 & 3 & \lambda \end{bmatrix}, \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \mathbf{b} = \begin{bmatrix} 5 \\ 9 \\ \mu \end{bmatrix}\)

Step 2: Identifying Infinite Solutions.

An infinite solution set occurs when the equations within the system are linearly dependent, implying that the rank of matrix \( A \) is less than the number of variables (which is 3). A straightforward method is to check if any equation is a scalar multiple or a linear combination of others.

Consider the substitution of \(\lambda = 1\) and \(\mu = 13\) into the system:

• \( x + y + z = 5 \)
• \( x + 2y + 1^2z = 9 \rightarrow x + 2y + z = 9 \)
• \( x + 3y + 1z = 13 \rightarrow x + 3y + z = 13 \)

This specific substitution simplifies the system into a state of linear dependency among the equations. Consequently, unique values for each variable cannot be determined, leading to an infinite number of solutions.

Step 3: Detecting Inconsistency.

An inconsistent system is one that has no solution. This situation arises if, during the process of row reduction, an equation of the form \( 0 = c \) emerges, where \( c eq 0 \).

When \(\lambda = 1\) and \(\mu eq 13\), the system will reduce to an inconsistent state. Altering the constant term \(\mu\) will result in a contradiction after attempting to solve the system.

Step 4: Consistency Check for \(\lambda eq 1\).

When \(\lambda eq 1\), the coefficient matrix does not exhibit linear dependency because \( \lambda^2 eq \lambda \). If \(\mu = 13\) under this condition, the system's consistency will be dependent on the specific value of \(\lambda\).

Step 5: Uniqueness Check for \(\lambda eq 1\) and \(\mu eq 13\).

If neither \(\lambda\) nor \(\mu\) causes the system to degenerate or present a contradiction, and both are varied, it implies a scenario where unique values for \( x, y, \) and \( z \) can be determined. This is because all equations can be uniquely solved to provide specific values for the variables.

Conclusion:

The statement "System has a unique solution if \(\lambda eq 1\) and \(\mu eq 13\)" is NOT accurate based on the preceding analysis.