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, we need to find the value of \(|\text{adj}\left(\frac{M}{2}\right)|\) given that \(|M| = 2\) and \(M\) is a square matrix of order 3.

### Step-by-Step Solution:

1. **Understanding the determinant of \(\frac{M}{2}\)**:
The determinant of a scalar multiple of a matrix can be expressed as:
\[ |kM| = k^n |M| \]
where \(n\) is the order of the matrix and \(k\) is the scalar. In this case, \(k = \frac{1}{2}\) and \(n = 3\).
Therefore,
\[ \left|\frac{M}{2}\right| = \left(\frac{1}{2}\right)^3 |M| = \frac{1}{8} \cdot 2 = \frac{1}{4} \]

2. **Finding the determinant of the adjoint of \(\frac{M}{2}\)**:
The determinant of the adjoint of a matrix \(X\) is given by:
\[ |\text{adj}(X)| = |X|^{n-1} \]
where \(n\) is the order of the matrix. Here, \(X = \frac{M}{2}\) and \(n = 3\).
Thus,
\[ |\text{adj}\left(\frac{M}{2}\right)| = \left|\frac{M}{2}\right|^{3-1} = \left|\frac{M}{2}\right|^{2} \]

3. **Substituting the value of \(|\frac{M}{2}|\)**:
We already calculated that \(\left|\frac{M}{2}\right| = \frac{1}{4}\). Now we substitute this value:
\[ |\text{adj}\left(\frac{M}{2}\right)| = \left(\frac{1}{4}\right)^{2} = \frac{1}{16} \]

### Final Answer:
Thus, the value of \(|\text{adj}\left(\frac{M}{2}\right)|\) is \(\frac{1}{16}\).

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