Step 1: Inconsistency Definition:
A linear system is inconsistent if it lacks a solution. This condition is met during Gaussian elimination when a contradictory equation of the form \(0 = k\), where \(k eq 0\), is derived. In matrix terms, this signifies that the rank of the coefficient matrix (A) is strictly less than the rank of the augmented matrix (A|B).
Step 2: Methodology:
The system will be represented by an augmented matrix and transformed into row echelon form using row operations.
\[ \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 6
1 & 4 & 3 & 10
1 & 4 & \lambda & \mu \end{array} \right] \]
Step 3: Execution:
Perform the following row operations:
1. \( R_2 \rightarrow R_2 - R_1 \)
2. \( R_3 \rightarrow R_3 - R_1 \)
\[ \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 6
0 & 2 & 2 & 4
0 & 2 & \lambda-1 & \mu-6 \end{array} \right] \]
Subsequently, apply \( R_3 \rightarrow R_3 - R_2 \):
\[ \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 6
0 & 2 & 2 & 4
0 & 0 & (\lambda-1)-2 & (\mu-6)-4 \end{array} \right] \]
\[ \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 6
0 & 2 & 2 & 4
0 & 0 & \lambda-3 & \mu-10 \end{array} \right] \]
The final row yields the equation \( (\lambda-3)z = \mu-10 \).
For the system to be inconsistent, this equation must present a contradiction, occurring when the left-hand side equals zero and the right-hand side is non-zero.
Requirement for zero left-hand side:
\[ \lambda - 3 = 0 \implies \lambda = 3 \]
Requirement for non-zero right-hand side:
\[ \mu - 10 eq 0 \implies \mu eq 10 \]
Step 4: Conclusion:
The system is inconsistent under the conditions \( \lambda = 3 \) and \( \mu eq 10 \). This aligns with option (C).