The rank of the matrix `[(1,2,3),(lambda , 2 , 4),(2,-3,1)]` is 3 if :

Options

A. `lambda ne 18/11`
B. `lambda ne 18/11`
C. `lambda =-18/11`
D. None of these

Answer

To determine the values of \( \lambda \) for which the rank of the matrix

\[ A = \begin{pmatrix} 1 & 2 & 3 \\ \lambda & 2 & 4 \\ 2 & -3 & 1 \end{pmatrix} \]

is 3, we need to ensure that the determinant of the matrix \( A \) is non-zero. If the determinant is non-zero, it implies that all three rows are linearly independent, which means the rank is 3.

### Step 1: Calculate the Determinant of Matrix A

The determinant of a 3x3 matrix

\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

is given by the formula:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix \( A \):

\[ \text{det}(A) = 1 \cdot (2 \cdot 1 - 4 \cdot (-3)) - 2 \cdot (\lambda \cdot 1 - 4 \cdot 2) + 3 \cdot (\lambda \cdot (-3) - 2 \cdot 2) \]

### Step 2: Simplify the Determinant Expression

Calculating each term:

1. The first term:
\[ 1 \cdot (2 \cdot 1 - 4 \cdot (-3)) = 1 \cdot (2 + 12) = 1 \cdot 14 = 14 \]

2. The second term:
\[ -2 \cdot (\lambda - 8) = -2\lambda + 16 \]

3. The third term:
\[ 3 \cdot (-3\lambda - 4) = -9\lambda - 12 \]

Putting it all together:

\[ \text{det}(A) = 14 - 2\lambda + 16 - 9\lambda - 12 \]

### Step 3: Combine Like Terms

Combining the constants and the \( \lambda \) terms:

\[ \text{det}(A) = (14 + 16 - 12) + (-2\lambda - 9\lambda) = 18 - 11\lambda \]

### Step 4: Set the Determinant Not Equal to Zero

To ensure the rank is 3, we need:

\[ 18 - 11\lambda \neq 0 \]

### Step 5: Solve for \( \lambda \)

Solving the equation:

\[ 18 \neq 11\lambda \]
\[ \lambda \neq \frac{18}{11} \]

### Final Result

The rank of the matrix \( A \) is 3 if \( \lambda \neq \frac{18}{11} \).