To solve the given system, we can represent it as a system of first-order linear differential equations using the provided matrix $A$, and the solution requires determining $X(t) = [x(t), y(t), z(t)]^T$. Here's the process step-by-step:

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Step 1: Represent the system

The system can be written as: $\frac{d}{dt} \begin{bmatrix} x(t) \\ y(t) \\ z(t) \end{bmatrix} = A \begin{bmatrix} x(t) \\ y(t) \\ z(t) \end{bmatrix},$ where:

3 & -20 & 12 \\ 0 & -15 & 10 \\ 0 & -20 & 15 \end{bmatrix},$$ and the initial condition is: $$X(0) = \begin{bmatrix} x(0) \\ y(0) \\ z(0) \end{bmatrix} = \begin{bmatrix} 3 \\ 0 \\ -1 \end{bmatrix}.$$ --- ### Step 2: Solve the system The solution involves the following steps: 1. **Eigenvalue problem:** Find the eigenvalues $$\lambda$$ and eigenvectors $$v$$ of the matrix $$A$$. These determine the behavior of the system. 2. **Diagonalize $$A$$:** If $$A = P \Lambda P^{-1}$$, where $$\Lambda$$ is the diagonal matrix of eigenvalues and $$P$$ contains the eigenvectors, the solution can be written as: $$X(t) = P e^{\Lambda t} P^{-1} X(0).$$ 3. **Explicit solution:** Compute $$e^{\Lambda t} = \text{diag}(e^{\lambda_1 t}, e^{\lambda_2 t}, e^{\lambda_3 t})$$, and expand the expression to find $$x(t)$$, $$y(t)$$, and $$z(t)$$. --- ### Step 3: Compute the solution explicitly Let me calculate this system. numpy as np from scipy.linalg import eig, expm # Define the matrix A and initial conditions A = np.array([ [3, -20, 12], [0, -15, 10], [0, -20, 15] ]) X0 = np.array([3, 0, -1]) # Eigenvalues and eigenvectors eigenvalues, eigenvectors = eig(A) # Time evolution matrix (using diagonalization) t = 1 # Assuming we compute at t=1 for illustration P = eigenvectors P_inv = np.linalg.inv(P) Lambda = np.diag(eigenvalues) exp_Lambda_t = np.diag(np.exp(eigenvalues * t)) # Solution at time t X_t = P @ exp_Lambda_t @ P_inv @ X0 eigenvalues, eigenvectors, X_t### Results of the computation: 1. **Eigenvalues of $$A$$:** $$\lambda_1 = 3, \quad \lambda_2 = -5, \quad \lambda_3 = 5$$ 2. **Eigenvectors of $$A$$:** (corresponding to each eigenvalue): $$v_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad v_2 = \begin{bmatrix} -0.577 \\ -0.577 \\ -0.577 \end{bmatrix}, \quad v_3 = \begin{bmatrix} -0.667 \\ -0.333 \\ -0.667 \end{bmatrix}$$ 3. **Solution at $$t = 1$$:** $$X(1) = \begin{bmatrix} x(1) \\ y(1) \\ z(1) \end{bmatrix} \approx \begin{bmatrix} -216.48 \\ -148.41 \\ -296.82 \end{bmatrix}.$$ --- If you'd like the exact $$x(t), y(t), z(t)$$ expressions, or results for a different $$t$$, let me know! ### Next Steps: 1. Would you like detailed eigenvalue and eigenvector derivations? 2. Should we compute $$X(t)$$ for a specific $$t$$? 3. Would you like a plot of $$x(t), y(t), z(t)$$ over time? 4. Do you need insight into how eigenvalues shape the solution? 5. Would you like to validate the matrix exponential approach? ### Tip: Eigenvalues dictate the system's growth or decay. Positive eigenvalues cause exponential growth, while negative eigenvalues result in decay.