Let `A` be the set of all `3xx3` skew-symmetri matrices whose entries are either `-1,0,or1.` If there are exactly three 0s three 1s, and there `(-1)' s` , then the number of such matrices is __________.

Explanation

To solve the problem of finding the number of 3x3 skew-symmetric matrices with specific entries, we can follow these steps:

### Step 1: Understand the properties of skew-symmetric matrices
A skew-symmetric matrix \( A \) has the property that \( A^T = -A \). This means that the diagonal elements must be zero, and the off-diagonal elements satisfy \( a_{ij} = -a_{ji} \).

### Step 2: Identify the structure of a 3x3 skew-symmetric matrix
A general 3x3 skew-symmetric matrix can be represented as:
\[ A = \begin{pmatrix} 0 & a_{12} & a_{13} \\ -a_{12} & 0 & a_{23} \\ -a_{13} & -a_{23} & 0 \end{pmatrix} \]
Here, \( a_{12}, a_{13}, \) and \( a_{23} \) are the off-diagonal elements.

### Step 3: Determine the entries of the matrix
According to the problem, we need to fill the matrix with exactly:
- 3 zeros
- 3 ones
- 3 minus ones

Since the diagonal elements are all zeros, we have to use the remaining entries (which are 3 off-diagonal positions) to fill with 1s and -1s.

### Step 4: Assign values to the off-diagonal elements
The off-diagonal elements \( a_{12}, a_{13}, \) and \( a_{23} \) can take values of either 1 or -1. Since we need to use all three 1s and three -1s, we can assign:
- 1 to two of the off-diagonal positions and -1 to the remaining one.

### Step 5: Count the arrangements
We can choose 2 positions out of the 3 available to place the 1s. The number of ways to choose 2 positions from 3 is given by the binomial coefficient:
\[ \binom{3}{2} = 3 \]
For each arrangement of 1s and -1s, we can permute the pairs of 1s and -1s in the matrix. Since each pair of off-diagonal elements is symmetric (i.e., \( a_{ij} = -a_{ji} \)), we have:
- For each arrangement of 1 and -1, there are \( 2! \) ways to arrange the two 1s and one -1.

### Step 6: Calculate the total number of matrices
Thus, the total number of skew-symmetric matrices can be calculated as:
\[ \text{Total Matrices} = \binom{3}{2} \times 2! = 3 \times 2 = 6 \]

### Final Answer
The number of such matrices is **6**.