Introduction:
We are given an n×n matrix A with entries 0 and n>1. It is stated that there is exactly one non-zero entry in each row and each column of matrix A. We need to determine the value of the determinant of matrix A.

Explanation:

1. Non-zero entries:
Since there is exactly one non-zero entry in each row and each column of matrix A, it implies that the non-zero entries in each row and each column must be distinct. Let's denote the non-zero entries in the ith row and jth column as a_ij.

2. Identifying the structure:
Let's consider the structure of matrix A. Since there are n rows and n columns, and each row and column has exactly one non-zero entry, it means that there is a total of n non-zero entries in the matrix.

3. Diagonal entries:
The diagonal entries of the matrix are the entries a_11, a_22, ..., a_nn. Since there is exactly one non-zero entry in each row and each column, it implies that the diagonal entries of matrix A must be the non-zero entries.

4. Permutation of rows and columns:
We can permute the rows and columns of the matrix A in any order without changing the determinant value. This is because the determinant of a matrix is invariant under row and column permutations.

5. Determinant of matrix A:
Since the diagonal entries of matrix A are the non-zero entries, the determinant of matrix A can be calculated as the product of these non-zero entries.

6. Determinant value:
As there are n non-zero entries in matrix A and each entry is equal to n, the determinant of matrix A can be calculated as det(A) = n * n * ... * n = n^n.

Conclusion:
Therefore, the determinant of matrix A is equal to n^n. In the given options, the correct answer is (C) n.