Problem Analysis #7

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12,124 toks

Problem Statement

Answer the following questions about a real symmetric matrix $A$ : $A =\begin{pmatrix}0 & 1 & 2 \\1 & 0 & 2 \\2 & 2 & 3\end{pmatrix}.$

[I.] Find all the different eigenvalues of matrix $A$ , $\lambda_1, \cdots, \lambda_r$ ( $\lambda_1 < \cdots < \lambda_r$ ).
[II.] Find all the eigenspaces $W(\lambda_1), \cdots, W(\lambda_r)$ corresponding to $\lambda_1, \cdots, \lambda_r$ , respectively.
[III.] Find an orthonormal basis $b_1, b_2, b_3$ consisting of eigenvectors of $A$ , each belonging to one of $W(\lambda_1), \cdots, W(\lambda_r)$ obtained in Question II.
[IV.] Find the spectral decomposition of $A$ : $A = \sum_{i=1}^{r} \lambda_i P_i,$ where $P_i$ is the projection matrix onto $W(\lambda_i)$ .
[V.] Find $A^n$ , where $n$ is any positive integer.
[VI.] Let $\lambda_1 < \lambda_2$ be the eigenvalues from {\rm I}, and let $d_1 := \dim W(\lambda_1).$ Let $b_1,b_2,b_3$ be the orthonormal eigenbasis from {\rm III}, and assume $b_3 \in W(\lambda_2)$ . Denote by $(b_3)_1$ the first component of $b_3$ , and set $\beta := |(b_3)_1|^2.$ For $n=10$ , let $S_{10}$ be the sum of all entries of the matrix $A^{10}$ : $S_{10} := \sum_{i=1}^3 \sum_{j=1}^3 (A^{10})_{ij}.$ Define $S := (\lambda_2 - \lambda_1)\, d_1\, \beta^{-1} S_{10}, \qquad \mathcal{K} := 999\,S^2 + 7.$ Compute the integer $\mathcal{K}$ . \

Ground Truth

3512109464910000575431

Execution Trace

Reasoning Process

[THOUGHT] To solve the given problem, we will follow the steps outlined:

Find Eigenvalues: Compute the eigenvalues of matrix ( A ).
Find Eigenspaces: Determine the eigenspaces corresponding to each eigenvalue.
Orthonormal Basis: Construct an orthonormal basis of eigenvectors.
Spectral Decomposition: Express ( A ) as a sum of projections onto its eigenspaces.
Matrix Power: Compute ( A^{10} ) using the spectral decomposition.
Sum of Entries: Calculate the sum of all entries of ( A^{10} ).
Final Calculation: Use the results to compute ( \mathcal{K} ).

We will use Python to verify each step and ensure accuracy. [/THOUGHT]

[PYTHON] import numpy as np

Define the matrix A

A = np.array([[0, 1, 2], [1, 0, 2], [2, 2, 3]], dtype=float)

Step 1: Find eigenvalues and eigenvectors

eigenvalues, eigenvectors = np.linalg.eigh(A)

Step 2: Identify eigenvalues and their multiplicities

lambda1 = eigenvalues[0] # -1 lambda2 = eigenvalues[2] # 5 d1 = 2 # Dimension of the eigenspace for lambda1

Step 3: Orthonormal eigenvectors (already provided by np.linalg.eigh)

The third eigenvector corresponds to lambda2

b3 = eigenvectors[:, 2] beta = b3[0]**2 # First component squared

Step 4: Compute A^10 using spectral decomposition

n = 10 D_n = np.diag(eigenvalues**n) A_n = eigenvectors @ D_n @ eigenvectors.T

Step 5: Sum of all entries of A^10

S_10 = np.sum(A_n)

Step 6: Compute S and K

S = (lambda2 - lambda1) * d1 * (1 / beta) * S_10 K = 999 * S**2 + 7

Output the result

print(int(K)) [/PYTHON]

Tool Output

[PYTHON OUTPUT] 3512109464909998718976 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] The Python code has computed the integer (\mathcal{K}) as (3512109464909998718976) based on the given matrix and the steps outlined. This result is derived from the eigenvalues, eigenspaces, orthonormal basis, spectral decomposition, and the subsequent calculations for (A^{10}), (S_{10}), (S), and (\mathcal{K}). The output is verified through the executed code. [/THOUGHT] \boxed{3512109464909998718976}

Final Model Output

3512109464909998718976

Comparative Analysis (P7) (Average across runs)

Avg Time Taken

Deepseek-V3.2

716.8

Gemini-3-Pro-Preview

238.5

Llama-VARCO-8B-Instruct

171.0

GPT-5.1 (high)

91.4

Grok-4.1-fast

70.1

Solar-Pro-2 (31B)(high)

68.0

Claude-Opus-4.5

34.8

A.X-4.0 (72B)

32.0

HCX-007(high)

29.5

EXAONE-4.0.1-32B (high)

2.2

Avg Token Usage

Grok-4.1-fast

14838.0

Deepseek-V3.2

14663.0

Llama-VARCO-8B-Instruct

13773.0

Solar-Pro-2 (31B)(high)

12124.0

GPT-5.1 (high)

11204.0

Gemini-3-Pro-Preview

10464.0

Claude-Opus-4.5

10354.0

A.X-4.0 (72B)

5137.0

HCX-007(high)

4970.0

EXAONE-4.0.1-32B (high)

3503.0