Problem Analysis #1

GPT-5.2 (high) · Run 1

34.30s
5,281 toks
Problem Statement

For a square matrix AA, the matrix exponential eAe^{A} is defined by

eA=E+∑k=1∞1k!Ak,e^{A} = E + \sum_{k=1}^{\infty} \frac{1}{k!}A^{k},

where EE is the identity matrix.

  • [(1)] Let
A=(−144−58103−3−5).A= \begin{pmatrix} -1 & 4 & 4 \\ -5 & 8 & 10 \\ 3 & -3 & -5 \end{pmatrix}. 
Compute the matrix exponential $e^{A}$.
  • [(2)] Consider the matrices
Ax=(0−x0x00001),B=(1000−10001),A_x= \begin{pmatrix} 0 & -x & 0 \\ x & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix},\qquad B= \begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, 
where $x$ is a real number, and define the vector
a=(11e).a= \begin{pmatrix} 1 \\[2mm] 1 \\[2mm] e \end{pmatrix}. 
Let $C = B e^{A_x}$.      Determine the minimum and maximum values of
yTCyyTy\frac{y^{T} C y}{y^{T}y} 
over all nonzero real vectors $y\in\mathbb{R}^{3}$.
  • [(3)] Let
f(z)=12zTCz−aTz,z=(z1z2z3),f(z)=\frac{1}{2}z^{T} C z - a^{T}z, \qquad z= \begin{pmatrix} z_{1}\\ z_{2}\\ z_{3} \end{pmatrix}, 
where $C$ is the matrix from part (2).      Find the value of
z12+z22+z32\sqrt{z_1^2+z_2^2+z_3^2} 
for the vector $z$ satisfying
∂f∂z1=∂f∂z2=∂f∂z3=0.\frac{\partial f}{\partial z_{1}} = \frac{\partial f}{\partial z_{2}} = \frac{\partial f}{\partial z_{3}} =0.

Let λ1,λ2,λ3\lambda_1,\lambda_2,\lambda_3 be the eigenvalues of the matrix AA in part {\rm (1)}. Let mm and MM be the minimum and maximum values, respectively, obtained in part {\rm (2)}, and let

R=z12+z22+z32R = \sqrt{z_1^2+z_2^2+z_3^2}

be the value obtained in part {\rm (3)}. Compute

λ1λ2λ3⋅m⋅R2⋅(λ12+λ22+λ32).\lambda_1\lambda_2\lambda_3 \cdot m \cdot R^2 \cdot \bigl(\lambda_1^2+\lambda_2^2+\lambda_3^2\bigr).

\

Ground Truth
252
Execution Trace
Reasoning Process

[THOUGHT] Compute the required scalar using invariants: λ1λ2λ3 = det(A) and λ1^2+λ2^2+λ3^2 = trace(A^2). From parts (2)-(3), compute C explicitly via exp(A_x), get its minimum Rayleigh quotient m and the critical-point z, hence R^2. Then multiply det(A)mR^2*trace(A^2). Verify all with Python exactly. [/THOUGHT] [PYTHON] import sympy as sp

A = sp.Matrix([[-1,4,4],[-5,8,10],[3,-3,-5]]) detA = sp.factor(A.det()) traceA2 = sp.factor((A**2).trace())

detA, traceA2 print("det(A) =", detA) print("trace(A^2) =", traceA2)

Parts (2)-(3)

x = sp.symbols('x', real=True) c, s = sp.cos(x), sp.sin(x) expAx = sp.Matrix([[c,-s,0],[s,c,0],[0,0,sp.E]]) B = sp.diag(1,-1,1) C = sp.simplify(B*expAx) print("C =", C)

eigenvalues of C (should be -1, 1, e)

eigsC = sp.factor(C.eigenvals()) print("eigenvals(C) =", eigsC)

m = min([sp.N(ev) for ev in eigsC.keys()]) # numeric for min; but we know -1 print("numeric min eigenvalue =", m)

exact min:

m_exact = sp.Integer(-1)

a = sp.Matrix([1,1,sp.E]) z = sp.simplify(C.LUsolve(a)) # solve Cz=a R2 = sp.simplify((z.T*z)[0]) print("z =", z) print("R^2 =", R2)

expr = sp.simplify(detA * m_exact * R2 * traceA2) print("final expression =", expr) [/PYTHON]

Tool Output
[PYTHON OUTPUT] det(A) = -6 trace(A^2) = 14 C = Matrix([[cos(x), -sin(x), 0], [-sin(x), -cos(x), 0], [0, 0, E]]) eigenvals(C) = {-1: 1, 1: 1, E: 1} numeric min eigenvalue = -1.00000000000000 z = Matrix([[sqrt(2)*cos(x + pi/4)], [-sqrt(2)*sin(x + pi/4)], [1]]) R^2 = 3 final expression = 252 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] Using the verified outputs: det⁡(A)=−6\det(A)=-6, tr⁡(A2)=14\operatorname{tr}(A^2)=14, m=−1m=-1, and R2=3R^2=3. Thus the requested product is (−6)(−1)(3)(14)=252(-6)(-1)(3)(14)=252.

252\boxed{252}
Final Model Output
252

Comparative Analysis (P1) (Average across runs)

Avg Time Taken

K-EXAONE-236B-A23B
575.7
GPT-5.1 (high)
150.0
Llama-VARCO-8B-Instruct
141.2
Deepseek-V3.2
124.1
A.X-4.0 (72B)
99.6
Gemini-3-Pro-Preview
97.2
Kanana-2-30B-Thinking
84.0
Kanana-2-30B-Thinking-2601
55.4
Claude-Opus-4.5
43.6
Solar-Pro-2 (31B)(high)
40.8
K-EXAONE-236B-A23B
40.2
HCX-007(high)
39.6
Grok-4.1-fast
34.8
GPT-5.2 (high)
34.3
Solar-Open-100B
32.5
EXAONE-4.0.1-32B (high)
6.3

Avg Token Usage

K-EXAONE-236B-A23B
59,097
Claude-Opus-4.5
19,293
Gemini-3-Pro-Preview
14,285
Kanana-2-30B-Thinking
13,812
Llama-VARCO-8B-Instruct
13,418
Kanana-2-30B-Thinking-2601
12,936
K-EXAONE-236B-A23B
11,127
Deepseek-V3.2
9,555
GPT-5.1 (high)
9,506
Grok-4.1-fast
9,320
Solar-Pro-2 (31B)(high)
8,677
Solar-Open-100B
8,267
HCX-007(high)
5,746
A.X-4.0 (72B)
5,358
GPT-5.2 (high)
5,281
EXAONE-4.0.1-32B (high)
5,236