If the matrix `[(1,3,lamda+2),(2,4,8),(3,5,10)]` is singular then `lamda=`

Options

A. `-2`
B. 4
C. 2
D. `-4`

Answer

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

\[ A = \begin{pmatrix} 1 & 3 & \lambda + 2 \\ 2 & 4 & 8 \\ 3 & 5 & 10 \end{pmatrix} \]

is singular, we need to calculate the determinant of the matrix and set it equal to zero.

### Step-by-Step Solution:

1. **Write the determinant of the matrix**:
The determinant of a \(3 \times 3\) matrix

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

is calculated as:

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

For our matrix \(A\), we have:
- \(a = 1\), \(b = 3\), \(c = \lambda + 2\)
- \(d = 2\), \(e = 4\), \(f = 8\)
- \(g = 3\), \(h = 5\), \(i = 10\)

Thus, the determinant can be expressed as:

\[ \text{det}(A) = 1(4 \cdot 10 - 8 \cdot 5) - 3(2 \cdot 10 - 8 \cdot 3) + (\lambda + 2)(2 \cdot 5 - 4 \cdot 3) \]

2. **Calculate each part of the determinant**:
- Calculate \(4 \cdot 10 - 8 \cdot 5\):
\[ 40 - 40 = 0 \]
- Calculate \(2 \cdot 10 - 8 \cdot 3\):
\[ 20 - 24 = -4 \]
- Calculate \(2 \cdot 5 - 4 \cdot 3\):
\[ 10 - 12 = -2 \]

3. **Substitute back into the determinant formula**:
Now substituting these values back, we get:

\[ \text{det}(A) = 1(0) - 3(-4) + (\lambda + 2)(-2) \]
Simplifying this gives:

\[ \text{det}(A) = 0 + 12 - 2(\lambda + 2) \]

4. **Set the determinant to zero**:
For the matrix to be singular, we set the determinant equal to zero:

\[ 12 - 2(\lambda + 2) = 0 \]

5. **Solve for \(\lambda\)**:
Expanding and solving the equation:

\[ 12 - 2\lambda - 4 = 0 \]
\[ 8 - 2\lambda = 0 \]
\[ 2\lambda = 8 \]
\[ \lambda = 4 \]

Thus, the value of \(\lambda\) for which the matrix is singular is

\[ \lambda = 4 \]