Let `A=[a_(ij)]` and `B=[b_(ij)]` be two `4xx4` real matrices such that `b_(ij)=(-2)^((i+j-2))a_(ji)` where `i,j=1,2,3,4.` If A is scalar matrix of determinant value of `1/128` then determinant of B is

Options

A. 64
B. `1/64`
C. 32
D. none of these

Explanation

To find the determinant of matrix \( B \) given the relationship between matrices \( A \) and \( B \), we can follow these steps:

### Step 1: Understand the relationship between \( A \) and \( B \)
We are given that:
\[ b_{ij} = (-2)^{(i+j-2)} a_{ji} \]
This means that each element of matrix \( B \) is derived from the elements of matrix \( A \) by multiplying \( a_{ji} \) with a power of \(-2\).

### Step 2: Write the determinant of \( B \)
The determinant of \( B \) can be expressed in terms of the determinant of \( A \):
\[ \text{det}(B) = \text{det} \begin{bmatrix} b_{11} & b_{12} & b_{13} & b_{14} \\ b_{21} & b_{22} & b_{23} & b_{24} \\ b_{31} & b_{32} & b_{33} & b_{34} \\ b_{41} & b_{42} & b_{43} & b_{44} \end{bmatrix} \]

### Step 3: Factor out the powers of \(-2\)
Using the relationship defined, we can factor out the powers of \(-2\) from each row of \( B \):
\[ \text{det}(B) = (-2)^{0} a_{11} \cdots + (-2)^{1} a_{12} \cdots + (-2)^{2} a_{13} \cdots + (-2)^{3} a_{14} \cdots \]
Continuing this for all rows, we can see that:
- The first row contributes \( (-2)^0 \)
- The second row contributes \( (-2)^1 \)
- The third row contributes \( (-2)^2 \)
- The fourth row contributes \( (-2)^3 \)

### Step 4: Calculate the total power of \(-2\)
The total power of \(-2\) that can be factored out is:
\[ (-2)^{0 + 1 + 2 + 3} = (-2)^{6} \]

### Step 5: Write the determinant in terms of \( A \)
Thus, we can write:
\[ \text{det}(B) = (-2)^{6} \cdot \text{det}(A) \]

### Step 6: Substitute the value of \( \text{det}(A) \)
We know that \( \text{det}(A) = \frac{1}{128} = \frac{1}{2^7} \). Therefore:
\[ \text{det}(B) = (-2)^{6} \cdot \frac{1}{2^7} \]

### Step 7: Simplify the expression
Calculating this gives:
\[ \text{det}(B) = \frac{(-2)^{6}}{2^7} = \frac{64}{128} = \frac{1}{2} \]

### Final Result
Thus, the determinant of matrix \( B \) is:
\[ \text{det}(B) = \frac{1}{2} \]