$n\geq 2$ is a given integer. For two permuations $(\alpha_1,\cdots,\alpha_n)$ and $(\beta_1,\cdots,\beta_n)$ of $1,\cdots,n$, consider $n\times n$ matrix $A= \left(a_{ij} \right)_{1\leq i,j\leq n}$ defined by $a_{ij} = (1+\alpha_i \beta_j )^{n-1}$. Find every possible value of $\det(A)$.
To find the determinant of the matrix \( A = \left(a_{ij}\right)_{1 \leq i,j \leq n} \) where \( a_{ij} = (1 + \alpha_i \beta_j)^{n-1} \), we can use properties of Vandermonde matrices. 

1. **Matrix Representation**:
   The matrix \( A \) can be represented as the product of two matrices:
   \[
   A = \left( (1 + \alpha_i \beta_j)^{n-1} \right)_{1 \leq i,j \leq n}
   \]
   We can express this as:
   \[
   A = \left( \sum_{k=0}^{n-1} \binom{n-1}{k} (\alpha_i \beta_j)^k \right)_{1 \leq i,j \leq n}
   \]
   This can be written as the product of two matrices:
   \[
   A = \left( \binom{n-1}{k} \alpha_i^k \right)_{1 \leq i \leq n, 0 \leq k \leq n-1} \cdot \left( \beta_j^k \right)_{0 \leq k \leq n-1, 1 \leq j \leq n}
   \]

2. **Matrix Decomposition**:
   Let \( B \) be the matrix with entries \( B_{ik} = \binom{n-1}{k} \alpha_i^k \) and \( C \) be the matrix with entries \( C_{kj} = \beta_j^k \). Then:
   \[
   A = B \cdot C
   \]
   The matrix \( B \) is a generalized Vandermonde matrix, and \( C \) is a Vandermonde matrix.

3. **Determinant of Vandermonde Matrices**:
   The determinant of a Vandermonde matrix \( V \) with entries \( V_{ij} = x_i^{j-1} \) is given by:
   \[
   \det(V) = \prod_{1 \leq i < j \leq n} (x_j - x_i)
   \]
   For the matrix \( B \), the determinant is:
   \[
   \det(B) = \prod_{1 \leq i < j \leq n} (\alpha_j - \alpha_i) \cdot \prod_{i=1}^n \binom{n-1}{0} \binom{n-1}{1} \cdots \binom{n-1}{n-1}
   \]
   Since \( \binom{n-1}{k} \) are constants, we have:
   \[
   \det(B) = \prod_{1 \leq i < j \leq n} (\alpha_j - \alpha_i) \cdot (n-1)!
   \]

4. **Determinant of Matrix \( C \)**:
   The matrix \( C \) is a standard Vandermonde matrix with entries \( C_{kj} = \beta_j^k \). Its determinant is:
   \[
   \det(C) = \prod_{1 \leq i < j \leq n} (\beta_j - \beta_i)
   \]

5. **Determinant of Matrix \( A \)**:
   Using the property of determinants for the product of matrices, we have:
   \[
   \det(A) = \det(B) \cdot \det(C)
   \]
   Substituting the determinants of \( B \) and \( C \):
   \[
   \det(A) = \left( \prod_{1 \leq i < j \leq n} (\alpha_j - \alpha_i) \cdot (n-1)! \right) \cdot \left( \prod_{1 \leq i < j \leq n} (\beta_j - \beta_i) \right)
   \]
   Simplifying, we get:
   \[
   \det(A) = (n-1)! \cdot \prod_{1 \leq i < j \leq n} (\alpha_j - \alpha_i) \cdot \prod_{1 \leq i < j \leq n} (\beta_j - \beta_i)
   \]

6. **Final Expression**:
   The determinant of \( A \) is:
   \[
   \boxed{\pm (n-1)!^n}
   \]
   The sign depends on the permutation of the elements \( \alpha_i \) and \( \beta_j \).