Word Count : 1084

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To solve the system of differential equations: \[ \frac{dy(t)}{dt} = A y(t), \quad y(0) = \begin{bmatrix}1\\1\\1\end{bmatrix}, \quad \text{where} \quad A = \begin{bmatrix}2 & -5 & -11\\0 & -2 & -9\\0 & 1 & 4\end{bmatrix}, \] we need to find the eigenvalues and eigenvectors of the matrix \( A \). Here's the step-by-step solution: --- ### Step 1: Find the eigenvalues of \( A \) The eigenvalues \( \lambda \) are found by solving the characteristic equation: \[ \det(A - \lambda I) = 0, \] where \( I \) is the identity matrix. Substituting \( A \): \[ A - \lambda I = \begin{bmatrix}2 - \lambda & -5 & -11\\0 & -2 - \lambda & -9\\0 & 1 & 4 - \lambda\end{bmatrix}. \] The determinant is: \[ \det(A - \lambda I) = (2 - \lambda) \cdot \det\begin{bmatrix}-2 - \lambda & -9\\1 & 4 - \lambda\end{bmatrix}. \] Compute the \( 2 \times 2 \) determinant: \[ \det\begin{bmatrix}-2 - \lambda & -9\\1 & 4 - \lambda\end{bmatrix} = (-2 - \lambda)(4 - \lambda) - (-9)(1) = (-2 - \lambda)(4 - \lambda) + 9. \] ________ ___ ______ ____ _________.
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