Let A and B are two matrices of order `3xx3`, where `|A|=-2 and |B|=2`, then `|A^(-1)adj(B^(-1))adj(2A^(-1))|` is equal to

Options

A. 2
B. `-2`
C. 4
D. 8

Answer

To solve the problem, we need to find the determinant of the expression \( |A^{-1} \text{adj}(B^{-1}) \text{adj}(2A^{-1})| \).

### Step-by-Step Solution:

1. **Use the determinant property of the inverse of a matrix**:
\[ |A^{-1}| = \frac{1}{|A|} \]
Given \( |A| = -2 \), we have:
\[ |A^{-1}| = \frac{1}{-2} = -\frac{1}{2} \]

2. **Use the determinant property of the adjugate of a matrix**:
\[ |\text{adj}(B)| = |B|^{n-1} \]
where \( n \) is the order of the matrix. Since \( B \) is a \( 3 \times 3 \) matrix, we have \( n = 3 \):
\[ |\text{adj}(B^{-1})| = |B^{-1}|^{2} = \left(\frac{1}{|B|}\right)^{2} \]
Given \( |B| = 2 \), we find:
\[ |B^{-1}| = \frac{1}{2} \implies |\text{adj}(B^{-1})| = \left(\frac{1}{2}\right)^{2} = \frac{1}{4} \]

3. **Calculate the determinant of the adjugate of \( 2A^{-1} \)**:
\[ |\text{adj}(kA)| = k^{n-1} |A|^{n-1} \]
For \( k = 2 \) and \( n = 3 \):
\[ |\text{adj}(2A^{-1})| = 2^{2} |\text{adj}(A^{-1})| = 4 |A^{-1}|^{2} = 4 \left(-\frac{1}{2}\right)^{2} = 4 \cdot \frac{1}{4} = 1 \]

4. **Combine the determinants**:
Now we can combine all the determinants:
\[ |A^{-1} \text{adj}(B^{-1}) \text{adj}(2A^{-1})| = |A^{-1}| \cdot |\text{adj}(B^{-1})| \cdot |\text{adj}(2A^{-1})| \]
Substituting the values we found:
\[ = \left(-\frac{1}{2}\right) \cdot \left(\frac{1}{4}\right) \cdot 1 \]
\[ = -\frac{1}{2} \cdot \frac{1}{4} = -\frac{1}{8} \]

### Final Result:
Thus, the value of \( |A^{-1} \text{adj}(B^{-1}) \text{adj}(2A^{-1})| \) is:
\[ -\frac{1}{8} \]