Let `A=(a_(ij))_(3xx3) and B=(b_(ij))_(3xx3)`, where `b_(ij)=(a_(ij)+a_(ji))/(2) Aai, j`. Number of such matrices A whose elements are selected from the set `{0, 1, 2, 3}` such that `A=B`. Are

Options

A. `2^(9)`
B. `2^(12)`
C. `2^(6)`
D. `2^(8)`

Answer

To solve the problem, we need to determine the number of symmetric matrices \( A \) of order \( 3 \times 3 \) such that \( A = B \), where \( B \) is defined as:

\[ b_{ij} = \frac{a_{ij} + a_{ji}}{2} \]

Given that the elements of matrix \( A \) are selected from the set \( \{0, 1, 2, 3\} \).

### Step-by-Step Solution:

1. **Understanding the Symmetric Matrix**:
For \( A \) to equal \( B \), we need:
\[ a_{ij} = b_{ij} = \frac{a_{ij} + a_{ji}}{2} \]
This equality holds if and only if \( a_{ij} = a_{ji} \). Thus, \( A \) must be a symmetric matrix.

2. **Identifying Elements of a \( 3 \times 3 \) Matrix**:
A \( 3 \times 3 \) symmetric matrix has the following structure:
\[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{13} & a_{23} & a_{33} \end{pmatrix} \]
Here, \( a_{12} = a_{21} \), \( a_{13} = a_{31} \), and \( a_{23} = a_{32} \).

3. **Counting Independent Elements**:
The independent elements of the symmetric matrix \( A \) are:
- \( a_{11} \)
- \( a_{12} \)
- \( a_{13} \)
- \( a_{22} \)
- \( a_{23} \)
- \( a_{33} \)

This gives us a total of 6 independent elements.

4. **Choosing Values for Each Element**:
Each element can be chosen from the set \( \{0, 1, 2, 3\} \), which contains 4 options. Therefore, for each of the 6 independent elements, we have 4 choices.

5. **Calculating Total Matrices**:
The total number of symmetric matrices \( A \) is given by:
\[ \text{Total Matrices} = 4^6 \]

6. **Simplifying the Expression**:
We can express \( 4^6 \) as:
\[ 4^6 = (2^2)^6 = 2^{12} \]

### Final Answer:
Thus, the number of such matrices \( A \) is \( 2^{12} \).