Find the number of all possible matrices of order `3xx3` with each entry 0 or 1. How many of these are symmetric ?

Explanation

To solve the problem of finding the number of all possible matrices of order \(3 \times 3\) with each entry being either 0 or 1, and then determining how many of these matrices are symmetric, we can follow these steps:

### Step 1: Calculate the total number of \(3 \times 3\) matrices
A \(3 \times 3\) matrix has 9 entries. Since each entry can either be 0 or 1, we can use the multiplication principle to find the total number of matrices.

\[ \text{Total matrices} = 2^{\text{number of entries}} = 2^9 = 512 \]

### Step 2: Determine the conditions for a matrix to be symmetric
A matrix \(A\) is symmetric if \(A = A^T\). This means that the elements across the main diagonal must be equal. In a \(3 \times 3\) matrix, the elements are arranged as follows:

\[ \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \]

For the matrix to be symmetric, the following conditions must hold:
- \(a_{12} = a_{21}\)
- \(a_{13} = a_{31}\)
- \(a_{23} = a_{32}\)

### Step 3: Count the independent entries in a symmetric matrix
In a symmetric \(3 \times 3\) matrix, the independent entries are:
- The diagonal elements: \(a_{11}, a_{22}, a_{33}\) (3 entries)
- The upper triangular elements: \(a_{12}, a_{13}, a_{23}\) (3 entries)

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

### Step 4: Calculate the number of symmetric matrices
Since each of the 6 independent entries can either be 0 or 1, we again use the multiplication principle:

\[ \text{Total symmetric matrices} = 2^{\text{number of independent entries}} = 2^6 = 64 \]

### Final Answer
- Total number of possible \(3 \times 3\) matrices: **512**
- Total number of symmetric \(3 \times 3\) matrices: **64**

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