[THOUGHT] I need to calculate the determinant of a 10×10 matrix where all diagonal entries are 20 and all off-diagonal entries are 10. This matrix can be written as A = 10J + 10I where J is the all-ones matrix and I is the identity matrix. Let me compute this determinant using Python. [/THOUGHT] [PYTHON] import numpy as np

Create the 10x10 matrix

n = 10 A = np.full((n, n), 10) # All entries are 10 np.fill_diagonal(A, 20) # Diagonal entries are 20

Calculate determinant

det_A = np.linalg.det(A) print(f"Determinant (float): {det_A}") print(f"Determinant (rounded): {round(det_A)}") [/PYTHON]

[THOUGHT] The determinant of the matrix A is 110000000000. This can also be verified analytically: the matrix A = 10I + 10J where J is the all-ones matrix. The eigenvalues of J are n=10 (with multiplicity 1) and 0 (with multiplicity n-1=9). So the eigenvalues of A are 10 + 10×10 = 110 (multiplicity 1) and 10 + 10×0 = 10 (multiplicity 9). The determinant is 110 × 10^9 = 110000000000. [/THOUGHT]

\boxed{110000000000}