Question

The augmented matrix of a non-homogeneous system of equations \( AX = B \) is reduced to the following form after applying a series of elementary row transformations: \[ \begin{bmatrix}1&1&1&5\\ 0&0&-3&4\\ 0&0&\mu+1&\lambda^{2}-2\lambda+1\end{bmatrix} \] Then, which of the following is correct?

• A. Only for \( \mu \neq -1 \), \( AX=B \) has a unique solution
• B. Only for \( \mu = -1 \) and \( \lambda = 1 \), \( AX=B \) has an infinite number of solutions
• C. For any value of \( \mu \) and \( \lambda \), \( AX=B \) has an infinite number of solutions
• D. For all positive values of \( \mu \), \( AX=B \) has no solution

Hint:

When a column of the coefficient matrix contains no leading pivot entry (like the second column here), the system can never have a unique solution, regardless of the parameters chosen.

Answer 1

The Correct Option is B

Concept: According to the Rouché-Capelli theorem for a system of linear equations with augmented matrix \( [A|B] \):

• Infinite Solutions: Occur when \( \text{Rank}(A) = \text{Rank}([A|B]) < \text{number of variables} \).

• No Solution (Inconsistent): Occurs when \( \text{Rank}(A) \neq \text{Rank}([A|B]) \).

• Unique Solution: Occurs when \( \text{Rank}(A) = \text{Rank}([A|B]) = \text{number of variables} \).

Step 1: Analyzing the Rank of matrix \( A \) and augmented matrix \( [A|B] \).
The reduced matrix is: \[ [A|B] = \begin{bmatrix}1 & 1 & 1 & \big| & 5 \\ 0 & 0 & -3 & \big| & 4 \\ 0 & 0 & \mu+1 & \big| & (\lambda-1)^2 \end{bmatrix} \] Let us look at the third row. For the system to have an infinite number of solutions, the entire last row must become zero so that the total rank drops below the number of variables (3). This requires: \[ \mu + 1 = 0 \implies \mu = -1 \] \[ (\lambda - 1)^2 = 0 \implies \lambda = 1 \]

Step 2: Checking option accuracy.
If \( \mu = -1 \) and \( \lambda = 1 \), the third row is entirely zero. Then \( \text{Rank}(A) = 2 \) and \( \text{Rank}([A|B]) = 2 \). Since the rank (2) is less than the number of variables (3), the system is consistent with infinitely many solutions. This precisely matches option (B). Note that a unique solution is impossible because column 2 has no pivot element.