Let `A=[a_(ij)]` and `B=[b_(ij)]` be two `3xx3` real matrices such that `b_(ij)=(3)(i+j-2)a_(ji)`, where i, j = 1,2,3. If the determinant of B is 81, then the determinant of A is :

Options

A. `1//9`
B. `1//81`
C. `1//3`
D. 3

Answer

To solve the problem, we need to find the determinant of matrix \( A \) given the relationship between matrices \( A \) and \( B \) and the determinant of \( B \).

### Step-by-Step Solution:

1. **Understanding the relationship**: We are given that \( b_{ij} = 3(i + j - 2)a_{ji} \) for \( i, j = 1, 2, 3 \). This means that each element of matrix \( B \) is dependent on the elements of matrix \( A \).

2. **Constructing matrix \( B \)**: We can express the elements of matrix \( B \) using the formula provided:
- For \( i = 1, j = 1 \): \( b_{11} = 3(1 + 1 - 2)a_{11} = 3^0 a_{11} = a_{11} \)
- For \( i = 1, j = 2 \): \( b_{12} = 3(1 + 2 - 2)a_{21} = 3^1 a_{21} = 3a_{21} \)
- For \( i = 1, j = 3 \): \( b_{13} = 3(1 + 3 - 2)a_{31} = 3^2 a_{31} = 9a_{31} \)
- For \( i = 2, j = 1 \): \( b_{21} = 3(2 + 1 - 2)a_{12} = 3^1 a_{12} = 3a_{12} \)
- For \( i = 2, j = 2 \): \( b_{22} = 3(2 + 2 - 2)a_{22} = 3^2 a_{22} = 9a_{22} \)
- For \( i = 2, j = 3 \): \( b_{23} = 3(2 + 3 - 2)a_{32} = 3^3 a_{32} = 27a_{32} \)
- For \( i = 3, j = 1 \): \( b_{31} = 3(3 + 1 - 2)a_{13} = 3^2 a_{13} = 9a_{13} \)
- For \( i = 3, j = 2 \): \( b_{32} = 3(3 + 2 - 2)a_{23} = 3^3 a_{23} = 27a_{23} \)
- For \( i = 3, j = 3 \): \( b_{33} = 3(3 + 3 - 2)a_{33} = 3^4 a_{33} = 81a_{33} \)

Thus, we can write matrix \( B \) as:
\[ B = \begin{bmatrix} a_{11} & 3a_{21} & 9a_{31} \\ 3a_{12} & 9a_{22} & 27a_{32} \\ 9a_{13} & 27a_{23} & 81a_{33} \end{bmatrix} \]

3. **Factoring out constants**: We can factor out the common terms from each column:
- From the first column, we can take out \( 1 \).
- From the second column, we can take out \( 3 \).
- From the third column, we can take out \( 9 \).

This gives us:
\[ B = 1 \cdot 3 \cdot 9 \cdot \begin{bmatrix} a_{11} & a_{21} & a_{31} \\ a_{12} & a_{22} & a_{32} \\ a_{13} & a_{23} & a_{33} \end{bmatrix} = 27A \]

4. **Finding the determinant**: The determinant of \( B \) can be expressed as:
\[ \text{det}(B) = 27 \cdot \text{det}(A) \]

5. **Using the given information**: We know that \( \text{det}(B) = 81 \). Therefore, we can set up the equation:
\[ 27 \cdot \text{det}(A) = 81 \]

6. **Solving for \( \text{det}(A) \)**:
\[ \text{det}(A) = \frac{81}{27} = 3 \]

### Final Answer:
The determinant of matrix \( A \) is \( 3 \).