[THOUGHT] The matrix A is 10 times the all-ones matrix plus 10 times the identity matrix. Its eigenvalues are 110 (multiplicity 1) and 10 (multiplicity 9), so det(A) = 110 * 10**9. To verify exactly, use sympy to construct the matrix and compute its determinant. [/THOUGHT] [PYTHON] from sympy import Matrix

Create the 10x10 matrix A

n = 10 A = Matrix(n, n, lambda i, j: 20 if i == j else 10)

Compute determinant

det_A = A.det() print(det_A) [/PYTHON]

[THOUGHT] The Python output confirms det(A) = 110000000000. This is the exact integer value. [/THOUGHT] \boxed{110000000000}

Explanation

The reasoning begins by recognizing the structure of matrix A: it consists of 20s on the diagonal and 10s elsewhere, equivalent to 10 times the all-ones matrix plus 10 times the 10x10 identity matrix. For such a matrix, the eigenvalues are derived as follows: the all-ones vector is an eigenvector with eigenvalue 1010 + 101 = 110 (due to the row sum), corresponding to multiplicity 1; the remaining eigenvalues are 10 (from the identity part), with multiplicity 9 for the orthogonal complement. Thus, the determinant equals the product of eigenvalues: 110 × 10^9 = 110000000000. To verify precisely, SymPy constructs the matrix using a lambda function (20 on diagonal, 10 off-diagonal) and computes the determinant, yielding the output 110000000000, confirming the analytical result.