If M be a square matrix of order 3 such that `|M|=2`, then `|adj((M)/(2))|` equals to :

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. `(1)/(8)`
D. `(1)/(16)`

Answer

To solve the problem step by step, we will follow the mathematical principles related to determinants and adjoint matrices.

### Step-by-Step Solution:

1. **Given Information:**
- We have a square matrix \( M \) of order 3, i.e., \( N = 3 \).
- The determinant of \( M \) is given as \( |M| = 2 \).

2. **Finding the Determinant of \( \frac{M}{2} \):**
- The formula for the determinant of a scalar multiple of a matrix is:
\[ |cA| = c^N |A| \]
where \( c \) is a scalar and \( A \) is a matrix of order \( N \).
- In our case, we have:
\[ | \frac{M}{2} | = \left( \frac{1}{2} \right)^N |M| \]
- Substituting \( N = 3 \) and \( |M| = 2 \):
\[ | \frac{M}{2} | = \left( \frac{1}{2} \right)^3 \cdot 2 = \frac{1}{8} \cdot 2 = \frac{1}{4} \]

3. **Finding the Determinant of the Adjoint of \( \frac{M}{2} \):**
- The determinant of the adjoint of a matrix \( A \) is given by:
\[ | \text{adj}(A) | = |A|^{N-1} \]
- Here, we need to find \( | \text{adj}(\frac{M}{2}) | \):
\[ | \text{adj}(\frac{M}{2}) | = | \frac{M}{2} |^{N-1} \]
- We already found \( | \frac{M}{2} | = \frac{1}{4} \) and \( N - 1 = 3 - 1 = 2 \):
\[ | \text{adj}(\frac{M}{2}) | = \left( \frac{1}{4} \right)^2 = \frac{1}{16} \]

4. **Final Answer:**
- Thus, the value of \( | \text{adj}(\frac{M}{2}) | \) is \( \frac{1}{16} \).

### Conclusion:
The final answer is:
\[ | \text{adj}(\frac{M}{2}) | = \frac{1}{16} \]