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

Answer: (D)

System has a unique solution if λ ≠ 1 and μ ≠ 13

Answer 2

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 $\mu \neq 13$.