Question

Determine the value of $\lambda$ and $\mu$ for which the system of equations
$x + 2y + z = 6$,
$x + 4y + 3z = 10$,
$2x + 4y + \lambda z = \mu$
has a unique solution.

• A. $\lambda = 2$, $\mu = 12$
• B. $\lambda = 2$, $\mu \neq 12$
• C. $\lambda \neq 2$, $\mu \neq 12$
• D. for any $\lambda$ and any $\mu$

Hint:

Ensure full rank for unique solutions in a system: avoid zero rows in coefficient matrix unless augmented part also zero.

Answer 1

The Correct Option is C

Step 1: Write augmented matrix
\[ \begin{bmatrix}[ccc|c] 1 & 2 & 1 & 6 \\ 1 & 4 & 3 & 10 \\ 2 & 4 & \lambda & \mu \end{bmatrix} \]
Step 2: Apply row operations to get row-echelon form
Subtract R₁ from R₂:
\[ R_2 \rightarrow R_2 - R_1 = [0, 2, 2, 4] \]
Subtract \( 2 \times R_1 \) from R₃:
\[ R_3 \rightarrow R_3 - 2R_1 = [0, 0, \lambda - 2, \mu - 12] \]
New matrix becomes:
\[ \begin{bmatrix}[ccc|c] 1 & 2 & 1 & 6 \\ 0 & 2 & 2 & 4 \\ 0 & 0 & \lambda - 2 & \mu - 12 \end{bmatrix} \]
Step 3: Conditions for unique solution
For a unique solution:
- The coefficient matrix must be of full rank.
- No row of the form \( [0\ 0\ 0\ |\text{nonzero}] \)
So, we must ensure:
\[ \lambda - 2 \neq 0 \Rightarrow \lambda \neq 2 \]
If \( \lambda = 2 \), then:
\[ \lambda - 2 = 0 \Rightarrow \text{row 3 becomes } [0\ 0\ 0\ |\mu - 12] \]
Now, if \( \mu \neq 12 \), it becomes inconsistent \( \Rightarrow \) no solution.
If \( \mu = 12 \), row becomes \( [0\ 0\ 0\ |0] \Rightarrow \) infinite solutions.
Hence, for unique solution:
\[ \boxed{\lambda \neq 2 \text{ and } \mu \neq 12} \]