Let A be the set of all `3xx3` symetric matrices whose entries are either 0 or 1. The number of elements is set A, is

Options

A. `2^3`
B. `2^6`
C. `2^9`
D. 18

Explanation

To find the number of elements in the set A of all \(3 \times 3\) symmetric matrices whose entries are either 0 or 1, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Structure of a Symmetric Matrix**:
A \(3 \times 3\) symmetric matrix has the following structure:
\[ \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{12} & a_{22} & a_{23} \\ a_{13} & a_{23} & a_{33} \end{pmatrix} \]
Here, the elements on the main diagonal are \(a_{11}, a_{22}, a_{33}\) and the elements above the diagonal are \(a_{12}, a_{13}, a_{23}\). The elements below the diagonal are determined by symmetry, i.e., \(a_{21} = a_{12}\), \(a_{31} = a_{13}\), and \(a_{32} = a_{23}\).

2. **Counting Independent Entries**:
In this symmetric matrix, we have:
- 3 independent entries on the diagonal: \(a_{11}, a_{22}, a_{33}\)
- 3 independent entries above the diagonal: \(a_{12}, a_{13}, a_{23}\)

Thus, there are a total of \(3 + 3 = 6\) independent entries.

3. **Choices for Each Entry**:
Each of these 6 independent entries can either be 0 or 1. Therefore, for each entry, there are 2 choices.

4. **Calculating the Total Number of Matrices**:
Since there are 6 independent entries and each can take 2 values, the total number of symmetric matrices can be calculated as:
\[ 2^6 \]

5. **Final Calculation**:
\[ 2^6 = 64 \]

Thus, the number of elements in set A is **64**.