The matrix `[(2,lambda,-4),(-1,3,4),(1,-2,-3)]` is non singular , if :

Options

A. `lambda ne -2`
B. `lambda ne 2 `
C. `lambda ne 3`
D. `lambda ne -3`

Answer

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

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

is non-singular, we need to find the determinant of the matrix and set it not equal to zero. A matrix is non-singular if its determinant is not equal to zero.

### Step 1: Calculate the Determinant

To find the determinant of the matrix \( A \), we can use the formula for the determinant of a \( 3 \times 3 \) matrix:

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

where \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \).

For our matrix:

- \( a = 2, b = \lambda, c = -4 \)
- \( d = -1, e = 3, f = 4 \)
- \( g = 1, h = -2, i = -3 \)

So, we can calculate:

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

### Step 2: Simplify Each Term

Now, we simplify each term:

1. Calculate \( 3 \cdot (-3) - 4 \cdot (-2) \):
\[ 3 \cdot (-3) = -9 \quad \text{and} \quad 4 \cdot (-2) = -8 \quad \Rightarrow \quad -9 + 8 = -1 \]

2. Calculate \( (-1) \cdot (-3) - 4 \cdot 1 \):
\[ (-1) \cdot (-3) = 3 \quad \text{and} \quad 4 \cdot 1 = 4 \quad \Rightarrow \quad 3 - 4 = -1 \]

3. Calculate \( (-1) \cdot (-2) - 3 \cdot 1 \):
\[ (-1) \cdot (-2) = 2 \quad \text{and} \quad 3 \cdot 1 = 3 \quad \Rightarrow \quad 2 - 3 = -1 \]

### Step 3: Substitute Back into the Determinant Formula

Now we substitute these results back into the determinant formula:

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

This simplifies to:

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

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

For the matrix to be non-singular, we need:

\[ \lambda + 2 \neq 0 \]

This leads to:

\[ \lambda \neq -2 \]

### Conclusion

Thus, the matrix is non-singular if \( \lambda \) is not equal to \(-2\).

### Final Answer

The correct option is:

**Option 1: \( \lambda \neq -2 \)**