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 conditions under which the given matrix 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.

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

### Step 1: Calculate the Determinant
We can calculate the determinant of the matrix \( A \) using the formula for the determinant of a 3x3 matrix:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is represented as:
\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

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

Now, substituting these values into the determinant formula:
\[ \text{det}(A) = 2(3 \cdot (-3) - 4 \cdot (-2)) - \lambda(-1 \cdot (-3) - 4 \cdot 1) - 4(-1 \cdot (-2) - 3 \cdot 1) \]

### Step 2: Simplify the Determinant
Calculating each part:
1. \( 3 \cdot (-3) = -9 \)
2. \( 4 \cdot (-2) = -8 \)
3. Therefore, \( 3 \cdot (-3) - 4 \cdot (-2) = -9 + 8 = -1 \)

Now substituting back:
\[ \text{det}(A) = 2(-1) - \lambda(3 - 4) - 4(2 - 3) \]

Calculating:
1. \( 2(-1) = -2 \)
2. \( 3 - 4 = -1 \)
3. \( -\lambda(-1) = \lambda \)
4. \( 2 - 3 = -1 \)
5. \( -4(-1) = 4 \)

Putting it all together:
\[ \text{det}(A) = -2 + \lambda + 4 = \lambda + 2 \]

### Step 3: Set the Determinant Not Equal to Zero
For the matrix to be non-singular, we need:
\[ \lambda + 2 \neq 0 \]
This implies:
\[ \lambda \neq -2 \]

### Conclusion
The matrix is non-singular if \( \lambda \) is not equal to -2.