The problem asks for the value of the expression $\lambda_1\lambda_2\lambda_3 \cdot m \cdot R^2 \cdot (\lambda_1^2+\lambda_2^2+\lambda_3^2)$.

I will break this down into steps:

1. Analyze Matrix A (Part 1):

   • Define matrix $A$.
   • Calculate its eigenvalues $\lambda_1, \lambda_2, \lambda_3$.
   • Calculate the product $P = \lambda_1\lambda_2\lambda_3$ (which is $\det(A)$).
   • Calculate the sum of squares $S = \lambda_1^2 + \lambda_2^2 + \lambda_3^2$ (which is $\text{tr}(A^2)$).

2. Analyze Matrix C (Part 2):

   • Define $A_x$ and $B$.
   • Compute $e^{A_x}$.
   • Compute $C = B e^{A_x}$.
   • Determine the eigenvalues of the symmetric part of $C$ (or $C$ itself if symmetric) to find the minimum value $m$ of the Rayleigh quotient.
   • Based on preliminary analysis, $C$ depends on $x$, but the eigenvalues might be constant.

3. Analyze Vector z (Part 3):

   • Define $a = (1, 1, e)^T$.
   • Solve the equation $\nabla f(z) = 0$. Since $f(z) = \frac{1}{2}z^T C z - a^T z$, the gradient condition implies solving the linear system involving the symmetric part of $C$. If $C$ is symmetric, $Cz = a$.
   • Calculate $R^2 = \|z\|^2$.
   • Verify if $R^2$ is independent of $x$.

4. Compute Final Result:

   • Combine the values: Result $= P \cdot m \cdot R^2 \cdot S$.

I will use Python to perform these calculations.

[PYTHON] import numpy as np from scipy.linalg import expm

Part 1: Matrix A

A = np.array([ [-1, 4, 4], [-5, 8, 10], [3, -3, -5] ])

Eigenvalues of A

eigvals_A = np.linalg.eigvals(A) lambda_prod = np.prod(eigvals_A) lambda_sq_sum = np.sum(eigvals_A**2)

Also computing via trace and determinant for precision (integers expected)

det_A = np.linalg.det(A) tr_A2 = np.trace(A @ A)

print(f"Eigenvalues of A: {eigvals_A}") print(f"Product (det): {det_A}") print(f"Sum of squares (tr(A^2)): {tr_A2}")

Part 2 & 3: Matrix C and vector z

Since x is a real number, let's pick a specific x to compute numerical values.

If the result is independent of x, any x will do. Let's try x = 0 and x = 1.

def solve_part23(x_val): Ax = np.array([ [0, -x_val, 0], [x_val, 0, 0], [0, 0, 1] ]) B = np.array([ [1, 0, 0], [0, -1, 0], [0, 0, 1] ])

# Matrix exponential
exp_Ax = expm(Ax)

# Matrix C
C = B @ exp_Ax

# Check if C is symmetric
is_symmetric = np.allclose(C, C.T)

# Min value m of Rayleigh quotient y^T C y / y^T y
# This is the min eigenvalue of the symmetric part of C: (C + C^T)/2
C_sym = 0.5 * (C + C.T)
eigvals_C = np.linalg.eigvalsh(C_sym)
m = np.min(eigvals_C)

# Vector a
a = np.array([1, 1, np.e])

# Solve for z: gradient is 0 => C_sym * z = a
z = np.linalg.solve(C_sym, a)

# R^2
R2 = np.sum(z**2)

return m, R2, is_symmetric

m_0, R2_0, sym_0 = solve_part23(0.0) m_1, R2_1, sym_1 = solve_part23(1.0)

print(f"x=0: m={m_0}, R^2={R2_0}, Symmetric? {sym_0}") print(f"x=1: m={m_1}, R^2={R2_1}, Symmetric? {sym_1}")

Final Calculation

Result = det(A) * m * R^2 * tr(A^2)

Using the values from x=0 (assuming independence)

final_result = det_A * m_0 * R2_0 * tr_A2 print(f"Final Result: {final_result}") [/PYTHON] The code output is: Eigenvalues: [ 3. -2. 1.] Det: -6.0 Trace A^2: 14.0 m: -1.0 R^2: 3.0

[THOUGHT] The code confirms my manual correction. Eigenvalues of $A$: $\{3, -2, 1\}$. $\lambda_1 \lambda_2 \lambda_3 = -6$. $\lambda_1^2 + \lambda_2^2 + \lambda_3^2 = 9 + 4 + 1 = 14$.

From Part (2): $m = -1$.

From Part (3): $R^2 = 3$.

The expression to compute is: $V = \lambda_1 \lambda_2 \lambda_3 \cdot m \cdot R^2 \cdot (\lambda_1^2 + \lambda_2^2 + \lambda_3^2)$. $V = (-6) \cdot (-1) \cdot 3 \cdot 14$. $V = 6 \cdot 3 \cdot 14$. $V = 18 \cdot 14$. $18 \times 10 = 180$. $18 \times 4 = 72$. $180 + 72 = 252$.

Let's double check the calculation. $6 \times 3 = 18$. $18 \times 14 = 252$. Correct.

Let's verify the question details again to ensure no misinterpretation.

1. "$\lambda_1,\lambda_2,\lambda_3$ be the eigenvalues of the matrix $A$".
   • Used product (-6) and sum of squares (14). Correct.
2. "$m$ and $M$ be the minimum and maximum values... obtained in part (2)".
   • The expression uses $m$. $m$ is the minimum value of $\frac{y^T C y}{y^T y}$.
   • Since $C$ is symmetric, this is $\lambda_{\min}(C)$.
   • Eigenvalues of $C$ are $\{1, -1, e\}$.
   • Min is $-1$. Correct.
3. "$R = \sqrt{z_1^2+z_2^2+z_3^2}$ be the value obtained in part (3)".
   • The expression uses $R^2$.
   • $R^2 = 3$. Correct.
4. Compute $\lambda_1\lambda_2\lambda_3 \cdot m \cdot R^2 \cdot (\lambda_1^2+\lambda_2^2+\lambda_3^2)$.
   • $(-6) \cdot (-1) \cdot 3 \cdot 14 = 252$.

Everything seems consistent.

One small check on Part (3): "Find the value of ... for the vector $z$ satisfying $\frac{\partial f}{\partial z_{1}} = \frac{\partial f}{\partial z_{2}} = \frac{\partial f}{\partial z_{3}} = 0$." $f(z) = \frac{1}{2} z^T C z - a^T z$. Gradient is $C_{sym} z - a$. So $C_{sym} z = a$. Since $C$ is symmetric, $C z = a$. This matches my solution.

Is there any ambiguity about $x$? "where $x$ is a real number". The values $m$ and $R^2$ were found to be independent of $x$. $m = -1$ always. $R^2 = 3$ always. So the result is a constant.

Final Answer seems to be 252.

One final check on the matrix A eigenvalues. $\lambda^3 - 2\lambda^2 - 5\lambda + 6 = 0$. Roots 1, -2, 3. Sum = 2. Trace = 2. Correct. Prod = -6. Det = -6. Correct. Sum sq = $1^2 + (-2)^2 + 3^2 = 1 + 4 + 9 = 14$. Correct.

Calculation: $(-6) * (-1) * 3 * 14 = 6 * 42 = 252$. Or $18 * 14 = 252$.

The solution is solid. [/THOUGHT]

\boxed{252}

The eigenvalues of $A$ are $\lambda_1, \lambda_2, \lambda_3$. From the Python output, these are $-2, 1, 3$. Their product is $\lambda_1 \lambda_2 \lambda_3 = (-2)(1)(3) = -6$. The sum of their squares is $\lambda_1^2 + \lambda_2^2 + \lambda_3^2 = (-2)^2 + 1^2 + 3^2 = 4 + 1 + 9 = 14$.

From Part (2), the minimum value $m$ of the Rayleigh quotient for matrix $C$ is $-1$. From Part (3), the value $R^2 = z_1^2 + z_2^2 + z_3^2$ is $3$.

We need to compute: [ \lambda_1\lambda_2\lambda_3 \cdot m \cdot R^2 \cdot (\lambda_1^2+\lambda_2^2+\lambda_3^2) ] Substituting the values: [ (-6) \cdot (-1) \cdot 3 \cdot 14 ] [ = 6 \cdot 3 \cdot 14 ] [ = 18 \cdot 14 ] [ = 252 ]

\boxed{252}