Problem Analysis #9

[THOUGHT] The given matrix A is a 10x10 symmetric matrix with diagonal entries 20 and off-diagonal entries 10. This matrix can be expressed as $A = 10I + 10J$, where $I$ is the identity matrix and $J$ is the matrix of all ones. The determinant of such a matrix is given by the formula $\det(A) = (a + nb) \cdot a^{n-1}$, where $a = 10$, $b = 10$, and $n = 10$. Applying the formula:

To verify, we will construct the matrix and compute its determinant using Python. [/THOUGHT] [PYTHON] import numpy as np

n = 10 A = np.full((n, n), 10) np.fill_diagonal(A, 20) det = np.linalg.det(A) print(round(det)) # Rounding to handle floating-point precision issues [/PYTHON]

[THOUGHT] The Python code output confirms the determinant of matrix $A$ is $110000000000$. This matches the analytical formula for matrices of the form $aI + bJ$, where $a = 10$, $b = 10$, and $n = 10$. The result is consistent and error-free. [/THOUGHT] $\boxed{110000000000}$