Since the sum of numbers in every row and every column is equal, we know that the sum of all the elements in the matrix is a multiple of 3. The sum of all the integers from 1 to 9 is 45, which is divisible by 3, so the sum of the elements in the matrix must be 3 times some integer.

Let's assume that the matrix has the following form:

$$
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
$$

We know that the sum of each row and each column is the same, so we can write three equations:

$$
a + b + c = d + e + f = g + h + i = S \quad\text{(sum of rows)}
$$

$$
a + d + g = b + e + h = c + f + i = S \quad\text{(sum of columns)}
$$

We can add these equations together to get:

$$
2S = (a + b + c + d + e + f + g + h + i) + (a + d + g + b + e + h + c + f + i)
$$

Simplifying this expression using the equations we have, we get:

$$
2S = 3S + (a + e + i)
$$

So, we can conclude that:

$$
(a + e + i) = S
$$

This means that the sum of the diagonal elements of the matrix is equal to the sum of each row and each column. Since all the elements in the matrix are distinct, we know that the diagonal elements are 3 distinct integers from 1 to 9 that add up to S.

We can start by choosing the largest integer for the middle element, since it will be part of all three diagonals. The only integer that works here is 9. Then, we can choose two distinct integers that add up to S-9 for the other two diagonal elements. There are two pairs of integers that work here: (1,8) and (2,7).

Once we have the diagonal elements, we can fill in the rest of the matrix by making sure that the sum of each row and each column is equal to S. We can start by filling in the middle row and middle column with the remaining integers (1,2,3,4,5,6,7,8) in any order, as long as the sum is equal to S-9. Then, we can fill in the top row and left column based on the values we have already filled in. Finally, we can fill in the bottom row and right column based on the values we have already filled in.

Here is one possible matrix that satisfies the conditions:

$$
\begin{pmatrix}
1 & 6 & 2 \\
7 & 9 & 3 \\
4 & 5 & 8
\end{pmatrix}
$$

In this matrix, the sum of each row and each column is 9, which is one third of the sum of all the integers from 1 to 9.