Assertion (A) : If the matrix `A=[{:(1,3,lambda+2),(2,4,8),(3,5,10):}]` is singular , then `lambda=4`.
Reason (R ) : If A is a singular matrix, then `|A|=0`

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false but R is true

Answer

To determine whether the assertion (A) is true and if it follows from the reason (R), we will analyze the matrix \( A \) and compute its determinant.

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

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

The determinant of a \( 3 \times 3 \) matrix can be calculated using the formula:
\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]
For our matrix \( A \), we have:
- \( a = 1, b = 3, c = \lambda + 2 \)
- \( d = 2, e = 4, f = 8 \)
- \( g = 3, h = 5, i = 10 \)

Substituting these values into the determinant formula:
\[ |A| = 1 \cdot (4 \cdot 10 - 8 \cdot 5) - 3 \cdot (2 \cdot 10 - 8 \cdot 3) + (\lambda + 2) \cdot (2 \cdot 5 - 4 \cdot 3) \]

### Step 2: Simplify Each Term

Calculating each term:
1. \( 4 \cdot 10 - 8 \cdot 5 = 40 - 40 = 0 \)
2. \( 2 \cdot 10 - 8 \cdot 3 = 20 - 24 = -4 \)
3. \( 2 \cdot 5 - 4 \cdot 3 = 10 - 12 = -2 \)

Substituting back into the determinant:
\[ |A| = 1 \cdot 0 - 3 \cdot (-4) + (\lambda + 2) \cdot (-2) \]
\[ |A| = 0 + 12 - 2(\lambda + 2) \]
\[ |A| = 12 - 2\lambda - 4 = 8 - 2\lambda \]

### Step 3: Set the Determinant to Zero for Singularity

For the matrix \( A \) to be singular:
\[ |A| = 0 \]
Thus, we set up the equation:
\[ 8 - 2\lambda = 0 \]

### Step 4: Solve for Lambda

Solving for \( \lambda \):
\[ 2\lambda = 8 \]
\[ \lambda = 4 \]

### Conclusion

The assertion (A) is true: If the matrix \( A \) is singular, then \( \lambda = 4 \). The reason (R) is also true, as a singular matrix has a determinant of zero.