Question

The system of equations
x + 2y + z = 6
x + 4y + 3z = 10
x + 4y + \(\lambda\)z = \(\mu\)
is inconsistent if

• A. \(\lambda \neq 3, \mu \in \mathbb{R}\)
• B. \(\lambda = 3, \mu = 10\)
• C. \(\lambda = 3, \mu \neq 10\)
• D. \(\lambda + \mu = 13\)

Hint:

For a system of linear equations to be inconsistent, you need a row in the echelon form of the augmented matrix that looks like \([0 \ 0 \ \ldots \ 0 \ | \ k]\) where \(k \neq 0\). This single condition encapsulates the requirement for no solution.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A system of linear equations is inconsistent if it has no solution. This occurs when, in the process of Gaussian elimination, we arrive at a contradictory equation of the form \(0 = k\), where \(k\) is a non-zero constant. In terms of matrix rank, this means the rank of the coefficient matrix (A) is less than the rank of the augmented matrix (A|B).
Step 2: Key Formula or Approach:
We will write the system as an augmented matrix and perform row operations to reduce it to row echelon form.
\[ \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 6
1 & 4 & 3 & 10
1 & 4 & \lambda & \mu \end{array} \right] \] Step 3: Detailed Explanation:
Apply the following row operations:
1. \( R_2 \rightarrow R_2 - R_1 \)
2. \( R_3 \rightarrow R_3 - R_1 \)
\[ \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 6
0 & 2 & 2 & 4
0 & 2 & \lambda-1 & \mu-6 \end{array} \right] \] Now, 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 last row represents the equation \( (\lambda-3)z = \mu-10 \).
For the system to be inconsistent (have no solution), this equation must be a contradiction. This happens when the left side is zero and the right side is non-zero.
Condition for the left side to be zero:
\[ \lambda - 3 = 0 \implies \lambda = 3 \] Condition for the right side to be non-zero:
\[ \mu - 10 \neq 0 \implies \mu \neq 10 \] Step 4: Final Answer:
The system is inconsistent if \( \lambda = 3 \) and \( \mu \neq 10 \). This corresponds to option (C).