Problem Analysis #7

16.23s

4,346 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] Compute eigenvalues/eigenvectors of the symmetric matrix using SymPy exactly. Since part VI assumes only two distinct eigenvalues, identify them (lambda1<lambda2) and d1=dim W(lambda1). Choose a normalized eigenvector b3 for lambda2, compute beta=|first component|^2, compute S10=sum of all entries of A^10, then form S and K=999*S^2+7 exactly as integers. [/THOUGHT] [PYTHON] import sympy as sp

A = sp.Matrix([[0,1,2], [1,0,2], [2,2,3]])

Eigenvalues (distinct) sorted

eigs = A.eigenvals() distinct = sorted(eigs.keys(), key=lambda x: sp.N(x)) print("eigenvals with mult:", eigs) print("distinct:", distinct)

Determine lambda1<lambda2 (assumes two distinct eigenvalues)

lambda1, lambda2 = distinct[0], distinct[-1] d1 = eigs[lambda1] print("lambda1, lambda2, d1:", lambda1, lambda2, d1)

Eigenvector for lambda2, normalized to get b3; choose first eigenvector from eigenvects()

evects = A.eigenvects() vec_lambda2 = None for val, mult, vects in evects: if sp.simplify(val - lambda2) == 0: vec_lambda2 = vects[0] break print("raw eigenvector for lambda2:", vec_lambda2)

Normalize

vec = sp.Matrix(vec_lambda2) norm2 = (vec.T*vec)[0] b3 = vec / sp.sqrt(norm2) beta = sp.simplify(b3[0]**2) # square since real; absolute value squared print("b3:", b3) print("beta:", beta)

Compute A^10 and S10

A10 = A**10 S10 = sum(A10) print("S10:", S10)

S = sp.simplify((lambda2 - lambda1) * d1 * S10 / beta) K = sp.simplify(999*S**2 + 7)

print("S:", S) print("K:", K)

Verify K is integer

print("K is integer?", sp.Integer(K) == K) print("K as int:", int(K)) [/PYTHON]

Tool Output

[PYTHON OUTPUT] eigenvals with mult: {5: 1, -1: 2} distinct: [-1, 5] lambda1, lambda2, d1: -1 5 2 raw eigenvector for lambda2: Matrix([[1/2], [1/2], [1]]) b3: Matrix([[sqrt(6)/6], [sqrt(6)/6], [sqrt(6)/3]]) beta: 1/6 S10: 26041667 S: 1875000024 K: 3512109464910000575431 K is integer? True K as int: 3512109464910000575431 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] Using the computed values, the required integer is (\mathcal{K}=999S^2+7) with (S=1875000024), giving: [ \boxed{3512109464910000575431} ]

Final Model Output

3512109464910000575431

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

axk1

13.1

EXAONE-4.0.1-32B (high)

2.2

Avg Token Usage

K-EXAONE-236B-A23B

294620.3

Solar-Open-100B

44634.0

Kanana-2-30B-Thinking

30454.0

K-EXAONE-236B-A23B

19396.0

Kanana-2-30B-Thinking-2601

17292.0

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

axk1

4390.0

GPT-5.2 (high)

4346.0

EXAONE-4.0.1-32B (high)

3503.0