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

Convert the system to matrix form and perform row reduction:

\[ \left(\begin{array}{ccc|c} 1 & 1 & 1 & 5 \\ 1 & 2 & 2 & 9 \\ 1 & 3 & \lambda & \mu \end{array}\right) \rightarrow \left(\begin{array}{ccc|c} 1 & 1 & 1 & 5 \\ 0 & 1 & 1 & 4 \\ 0 & 2 & \lambda - 1 & \mu - 5 \end{array}\right) \rightarrow \left(\begin{array}{ccc|c} 1 & 1 & 1 & 5 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & \lambda - 3 & \mu - 13 \end{array}\right) \]

For consistency: Unique solution if \(\lambda \neq 3\).

Infinite solutions if \(\lambda = 3\) and \(\mu = 13\).

No solution if \(\lambda = 3\) and \(\mu \neq 13\).

Therefore, the incorrect statement is:

(4) System has unique solution if \(\lambda = 1\) and \(\mu \neq 13\).