Problem Analysis #7

70.15s

14,838 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] First computation: define A and compute eigenvects to get eigenvalues and eigenvectors. [/THOUGHT] [PYTHON] from sympy import * A = Matrix([[0,1,2],[1,0,2],[2,2,3]]) ev = A.eigenvects() pprint(ev) [/PYTHON]

Tool Output

[PYTHON OUTPUT] ⎡⎛ ⎡⎡-1⎤ ⎡-2⎤⎤⎞ ⎛ ⎡⎡1/2⎤⎤⎞⎤ ⎢⎜ ⎢⎢ ⎥ ⎢ ⎥⎥⎟ ⎜ ⎢⎢ ⎥⎥⎟⎥ ⎢⎜-1, 2, ⎢⎢1 ⎥, ⎢0 ⎥⎥⎟, ⎜5, 1, ⎢⎢1/2⎥⎥⎟⎥ ⎢⎜ ⎢⎢ ⎥ ⎢ ⎥⎥⎟ ⎜ ⎢⎢ ⎥⎥⎟⎥ ⎣⎝ ⎣⎣0 ⎦ ⎣1 ⎦⎦⎠ ⎝ ⎣⎣ 1 ⎦⎦⎠⎦ [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] The eigenvalues are $\lambda_1 = -1$ and $\lambda_2 = 5$ . The eigenspace $W(\lambda_1)$ has dimension $d_1 = 2$ and is spanned by $\begin{pmatrix} -1 \\ 1 \\ 0 \end{pmatrix}$ and $\begin{pmatrix} -2 \\ 0 \\ 1 \end{pmatrix}$ . The eigenspace $W(\lambda_2)$ is spanned by $\begin{pmatrix} 1/2 \\ 1/2 \\ 1 \end{pmatrix}$ . An orthonormal eigenbasis can be obtained by Gram-Schmidt on the first eigenspace and normalizing the second eigenvector. The normalized $b_3 = \frac{1}{\sqrt{3/2}} \begin{pmatrix} 1/2 \\ 1/2 \\ 1 \end{pmatrix}$ , so $\beta = |(b_3)_1|^2 = 1/6$ . Then $S_{10} = 26041667$ , $S = 1875000024$ , and $\mathcal{K} = 3512109375089910575431$ . [/THOUGHT] \boxed{3512109375089910575431}

Explanation

The provided AI-generated answer derives from the spectral decomposition of the symmetric matrix (A) and targeted computations for part VI. The process begins with the eigenvalues and eigenvectors from the SymPy output: (\lambda_1 = -1) (algebraic multiplicity 2, geometric multiplicity 2, spanned by ([-1, 1, 0]^T) and ([-2, 0, 1]^T)) and (\lambda_2 = 5) (multiplicity 1, spanned by ([1/2, 1/2, 1]^T)).

For part VI, compute (\beta = |(b_3)_1|^2), where (b_3) is the unit eigenvector for (\lambda_2). The vector (u_3 = [1/2, 1/2, 1]^T) has norm (|u_3| = \sqrt{(1/2)^2 + (1/2)^2 + 1^2} = \sqrt{3/2}). Thus, (b_3 = u_3 / \sqrt{3/2}), and the first component is ((1/2) / \sqrt{3/2} = \sqrt{2/12} = 1/\sqrt{6}), so (\beta = (1/\sqrt{6})^2 = 1/6).

Next, (S_{10} = \sum_{i,j} (A^{10}){ij} = \mathbf{1}^T A^{10} \mathbf{1}), where (\mathbf{1} = [1, 1, 1]^T). Using spectral decomposition, (A^{10} = (-1)^{10} P_1 + 5^{10} P_2 = P_1 + 5^{10} P_2), so (S{10} = \mathbf{1}^T P_1 \mathbf{1} + 5^{10} \mathbf{1}^T P_2 \mathbf{1}). Here, (\mathbf{1}^T P_2 \mathbf{1} = (\mathbf{1}^T b_3)^2). Compute (\mathbf{1}^T u_3 = 2), so (\mathbf{1}^T b_3 = 2 / \sqrt{3/2} = 2 \sqrt{2/3}), and (\mathbf{1}^T P_2 \mathbf{1} = [2 \sqrt{2/3}]^2 = 8/3). Then (\mathbf{1}^T P_1 \mathbf{1} = \mathbf{1}^T \mathbf{1} - 8/3 = 3 - 8/3 = 1/3). With (5^{10} = 9,765,625), (S_{10} = 1/3 + (8/3) \cdot 9,765,625 = (1 + 78,125,000)/3 = 78,125,001 / 3 = 26,041,667).

Now, (S = (\lambda_2 - \lambda_1) d_1 \beta^{-1} S_{10} = (5 - (-1)) \cdot 2 \cdot 6 \cdot 26,041,667 = 6 \cdot 2 \cdot 6 \cdot 26,041,667 = 72 \cdot 26,041,667 = 1,875,000,024).

Finally, (\mathcal{K} = 999 S^2 + 7). Compute (S^2 = (1,875,000,024)^2 = 3,515,625,000,090,000,576). Then (999 S^2 = 1,000 S^2 - S^2 = 3,515,625,000,090,000,576,000 - 3,515,625,000,090,000,576 = 3,515,625,000,090,000,576,000 - 3,515,625,000,090,000,576) (via digit-wise subtraction with borrows), yielding (3,512,109,375,089,910,575,424). Adding 7 gives (\mathcal{K} = 3,512,109,375,089,910,575,431).

Final Model Output

3512109375089910575431

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