Problem Analysis #7

Solar-Pro-2 (31B)(high) · Run 1

68.01s
12,124 toks
Problem Statement

Answer the following questions about a real symmetric matrix AA:

A=(012102223).A =\begin{pmatrix}0 & 1 & 2 \\1 & 0 & 2 \\2 & 2 & 3\end{pmatrix}.
  • [I.] Find all the different eigenvalues of matrix AA, λ1,⋯ ,λr\lambda_1, \cdots, \lambda_r (λ1<⋯<λr\lambda_1 < \cdots < \lambda_r).
  • [II.] Find all the eigenspaces W(λ1),⋯ ,W(λr)W(\lambda_1), \cdots, W(\lambda_r) corresponding to λ1,⋯ ,λr\lambda_1, \cdots, \lambda_r, respectively.
  • [III.] Find an orthonormal basis b1,b2,b3b_1, b_2, b_3 consisting of eigenvectors of AA, each belonging to one of W(λ1),⋯ ,W(λr)W(\lambda_1), \cdots, W(\lambda_r) obtained in Question II.
  • [IV.] Find the spectral decomposition of AA:
A=∑i=1rλiPi,A = \sum_{i=1}^{r} \lambda_i P_i, 
where $P_i$ is the projection matrix onto $W(\lambda_i)$.
  • [V.] Find AnA^n, where nn is any positive integer.
  • [VI.] Let λ1<λ2\lambda_1 < \lambda_2 be the eigenvalues from {\rm I}, and let
d1:=dim⁡W(λ1).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
β:=∣(b3)1∣2.\beta := |(b_3)_1|^2. 
For $n=10$, let $S_{10}$ be the sum of all entries of the matrix $A^{10}$:
S10:=∑i=13∑j=13(A10)ij.S_{10} := \sum_{i=1}^3 \sum_{j=1}^3 (A^{10})_{ij}. 
Define
S:=(λ2−λ1) d1 β−1S10,K:=999 S2+7.S := (\lambda_2 - \lambda_1)\, d_1\, \beta^{-1} S_{10}, \qquad \mathcal{K} := 999\,S^2 + 7. 
Compute the integer $\mathcal{K}$.

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Ground Truth
3512109464910000575431
Execution Trace
Reasoning Process

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

  1. Find Eigenvalues: Compute the eigenvalues of matrix AA.
  2. Find Eigenspaces: Determine the eigenspaces corresponding to each eigenvalue.
  3. Orthonormal Basis: Construct an orthonormal basis of eigenvectors.
  4. Spectral Decomposition: Express AA as a sum of projections onto its eigenspaces.
  5. Matrix Power: Compute A10A^{10} using the spectral decomposition.
  6. Sum of Entries: Calculate the sum of all entries of A10A^{10}.
  7. Final Calculation: Use the results to compute K\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 K\mathcal{K} as 35121094649099987189763512109464909998718976 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 A10A^{10}, S10S_{10}, SS, and K\mathcal{K}. The output is verified through the executed code. [/THOUGHT] 3512109464909998718976\boxed{3512109464909998718976}

Final Model Output
3512109464909998718976

Comparative Analysis (P7) (Average across runs)

Avg Time Taken

K-EXAONE-236B-A23B
3160.3
Deepseek-V3.2
716.8
Solar-Open-100B
271.9
Gemini-3-Pro-Preview
238.5
Kanana-2-30B-Thinking
205.7
Llama-VARCO-8B-Instruct
171.0
K-EXAONE-236B-A23B
142.3
GPT-5.1 (high)
91.4
Kanana-2-30B-Thinking-2601
70.2
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
GPT-5.2 (high)
16.2
EXAONE-4.0.1-32B (high)
2.2

Avg Token Usage

K-EXAONE-236B-A23B
294,620
Solar-Open-100B
44,634
Kanana-2-30B-Thinking
30,454
K-EXAONE-236B-A23B
19,396
Kanana-2-30B-Thinking-2601
17,292
Grok-4.1-fast
14,838
Deepseek-V3.2
14,663
Llama-VARCO-8B-Instruct
13,773
Solar-Pro-2 (31B)(high)
12,124
GPT-5.1 (high)
11,204
Gemini-3-Pro-Preview
10,464
Claude-Opus-4.5
10,354
A.X-4.0 (72B)
5,137
HCX-007(high)
4,970
GPT-5.2 (high)
4,346
EXAONE-4.0.1-32B (high)
3,503