Question

Consider the system of linear equations $x + y + z = 4\mu$, $x + 2y + 2\lambda z = 10\mu$, $x + 3y + 4\lambda^2 z = \mu^2 + 15$, where $\lambda, \mu \in \mathbb{R}$. Which one of the following statements is NOT correct?

• A. The system has a unique solution if $\lambda \neq \frac{1}{2}$ and $\mu \neq 1, 15$.
• B. The system has an infinite number of solutions if $\lambda = \frac{1}{2}$ and $\mu = 15$.
• C. The system is inconsistent if $\lambda = \frac{1}{2}$ and $\mu \neq 1$.
• D. The system is consistent if $\lambda \neq \frac{1}{2}$.

Answer 1

Answer: (C)

The system is inconsistent if $\lambda = \frac{1}{2}$ and $\mu \neq 1$.

Answer 2

Write the system of equations in matrix form:

$\begin{bmatrix} 1 & 1 & 1 \\\ 1 & 2 & 2 \\\ 1 & 3 & 4\lambda \end{bmatrix} \begin{bmatrix} x \\\ y \\\ z \end{bmatrix} = \begin{bmatrix} 4\mu \\\ 10\mu \\\ \mu^2 + 15 \end{bmatrix}$

Let the coefficient matrix be $A$:

$A = \begin{bmatrix} 1 & 1 & 1 \\\ 1 & 2 & 2 \\\ 1 & 3 & 4\lambda \end{bmatrix}$

Calculate the determinant of $A$:

$\text{det}(A) = \begin{vmatrix} 1 & 1 & 1 \\\ 1 & 2 & 2 \\\ 1 & 3 & 4\lambda \end{vmatrix} = (2\lambda - 1)^2$

For unique solutions, $\text{det}(A) \neq 0$ or $\lambda \neq \frac{1}{2}$.

For infinite solutions, $\lambda = \frac{1}{2}$, and consistency depends on the rank of the augmented matrix with specific values of $\mu$.

The system is inconsistent if $\lambda = \frac{1}{2}$ and $\mu \neq 1$