Explanation:

Let's analyze the given matrix and conditions to determine the correct answer:

- You have a determinant that turns out to be zero for every \( a > 0 \).

- The given determinant is a special 3x3 matrix.

- For the determinant to be zero, the rows (or columns) must be linearly dependent.

Consider the determinant properties:

- If \( a, b, c \) are in Arithmetic Progression (AP), then \( b = \frac{a+c}{2} \).

- If \( a, b, c \) are in Geometric Progression (GP), then \( b^2 = ac \).

Now, analyze each option:

1. Option 1 (AP): The condition \( b = \frac{a+c}{2} \) means rows are not linearly dependent for all \( a > 0 \).

2. Option 2 (GP): With \( b^2 = ac \), you achieve the necessary linear dependency for all positive \( a \).

3. Option 3 (a, 2b, c in AP): This doesn't hold true as it does not satisfy the linear dependency.

4. Option 4 (a, 2b, c in GP): This condition doesn't naturally lead to zero determinant for all values of \( a > 0 \).

Conclusion:

- The correct answer is Option 2: a, b, c are in GP.