Problem Analysis #9

[THOUGHT] The matrix $A$ is a symmetric matrix of size $10 \times 10$. We can write it as [ A = 10 I + B ] where $I$ is the identity matrix and $B$ has all entries equal to 1 except for the diagonal which are zero. We can also write $A = 10 I + J$, where $J$ is a matrix with all entries equal to 1. The eigenvalues of $A$ are given by $\lambda_i = 10 + \mu_i$, where $\mu_i$ are the eigenvalues of $J$. The matrix $J$ has rank 1, so it has only one non-zero eigenvalue. The sum of the eigenvalues of $J$ is equal to its trace, which is 10. Since rank(J) = 1, there are 9 eigenvalues equal to zero and one eigenvalue equal to 10. Thus the eigenvalues of $J$ are $\mu_1 = 10, \mu_2 = \dots = \mu_{10} = 0$. The eigenvalues of $A$ are $\lambda_1 = 20, \lambda_2 = \dots = \lambda_{10} = 10$. The determinant of $A$ is the product of its eigenvalues. Thus, [ \det(A) = 20 \cdot 10^9 ]

Now we write the code to verify this. [PYTHON] import numpy as np

A = np.array([[20, 10, 10, 10, 10, 10, 10, 10, 10, 10], [10, 20, 10, 10, 10, 10, 10, 10, 10, 10], [10, 10, 20, 10, 10, 10, 10, 10, 10, 10], [10, 10, 10, 20, 10, 10, 10, 10, 10, 10], [10, 10, 10, 10, 20, 10, 10, 10, 10, 10], [10, 10, 10, 10, 10, 20, 10, 10, 10, 10], [10, 10, 10, 10, 10, 10, 20, 10, 10, 10], [10, 10, 10, 10, 10, 10, 10, 20, 10, 10], [10, 10, 10, 10, 10, 10, 10, 10, 20, 10], [10, 10, 10, 10, 10, 10, 10, 10, 10, 20]])

N = np.linalg.det(A) print(int(N))

[THOUGHT] The Python output gives the determinant of $A$ as 109999999999. This is close to $10^{10}$, but not exactly equal to it. The determinant of a matrix can be computed as the product of its eigenvalues. We found that the eigenvalues are 20 and nine times 10. Thus, $\det(A) = 20 \cdot 10^9$. However the Python output shows that it is actually $109999999999$. The calculation of the determinant is correct. The eigenvalues are 20,10,10,...10 (9 times). The determinant is $20 \times 10^9 = 2 \times 10^{10} = 20000000000$. The Python output is wrong.

Let's consider the matrix $A = 10I + J$, where $J$ is a matrix with all entries equal to 1. The eigenvalues of $J$ are $\lambda_i = n$, where $n$ is the size of the matrix. In this case, $\lambda_1 = 10$ and $\lambda_2 = \dots = \lambda_{10} = 0$. The eigenvalues of $A$ are $\mu_i = 10 + \lambda_i$. Thus, $\mu_1 = 20$ and $\mu_2 = \dots = \mu_{10} = 10$. The determinant of $A$ is $\det(A) = \prod_{i=1}^{10} \mu_i = 20 \cdot 10^9 = 2 \times 10^{10} = 20000000000$.

The Python output is incorrect. The correct answer should be 20000000000.

However, the Python output is given as 109999999999. This seems to be an error in the calculation.

The determinant is $20 \times 10^9 = 20,000,000,000$.

[PYTHON] import numpy as np

A = np.array([[20, 10, 10, 10, 10, 10, 10, 10, 10, 10], [10, 20, 10, 10, 10, 10, 10, 10, 10, 10], [10, 10, 20, 10, 10, 10, 10, 10, 10, 10], [10, 10, 10, 20, 10, 10, 10, 10, 10, 10], [10, 10, 10, 10, 20, 10, 10, 10, 10, 10], [10, 10, 10, 10, 10, 20, 10, 10, 10, 10], [10, 10, 10, 10, 10, 10, 20, 10, 10, 10], [10, 10, 10, 10, 10, 10, 10, 20, 10, 10], [10, 10, 10, 10, 10, 10, 10, 10, 20, 10], [10, 10, 10, 10, 10, 10, 10, 10, 10, 20]])

N = np.linalg.det(A) print(int(N))

[THOUGHT] The Python output consistently gives 109999999999. Despite my initial calculations suggesting a value of 20000000000, I must trust the Python output as ground truth.

Therefore, the determinant of A is 109999999999.

\boxed{109999999999}