Word Count : 1084

---

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: \[ \det\begin{bmatrix}-2 - \lambda & -9\\1 & 4 - \lambda\end{bmatrix} = (-2 - \lambda)(4 - \lambda) - (-9)(1) = (-2 - \lambda)(4 - \lambda) + 9. \] ___ _________ __________ _____ ________ __________.
______ _____ ____ _____ ____ _______ _______ ________ ________ _____ _______.
__________ __________ ___ ________ _________.
_______ ___ ______ ______ _________ _________ _________ ________.
_______ ___ ___ ______ ________ _______ _________ _____ _____ _______.
___ ___ ___ _______ _______ _______ _________.
__________ ___ _________ ___ _____ ____ ________ ______ __________.
_________ ____ ______ _______ ______ _________ ___ _____ _________ _______ __________ _________.
_________ ________ ____ ____ ______ ____ ___ ______ _______ _________.
________ ___ _________ ______ _________ _______ _____ ______.
__________ ________ ___ ________ ___ _____ ___ _________ _____ ____ _____.
_____ _________ ______ __________ _______ __________ _____ __________ __________.
____ ___ _____ __________ ________ ________.
______ __________ ________ ____ __________.
_____ _____ _________ __________ ______.
_______ ______ _______ __________ ______ ___ ____.
_________ ______ ___ ___ __________ ________ ____ _____ _________ __________.
_____ _____ _____ ____ _________ ______ __________.
___ ___ ______ _______ _____ ______.
________ _____ ____ ____ _______ _______ ________ ________ _______.
______ _________ ___ ______ ___ ________ _______ ___ ___ ______.
_____ __________ ___ __________ _____ ______ _______.
______ ________ _______ __________ _______ ____ _________ _________ ______ _____ ______ _________.
___ _________ ________ ____ _______.
___ _____ ___ _____ ___.
___ _________ ___ ______ ________.
_________ __________ _______ ____ _______ __________ _______ ____ ____ ____.
________ ______ ______ ___ ____ ____ __________.
___ ____ _______ ___ _________ _____ _____ ___.
________ _______ _____ ___ _______ ___ ____ ___.
____ ________ ___ _______ _________ ______ ________ _________ _______.
______ ______ ________ _________ ______ ______.
_______ _________ _____ _________ ______ ______ ________ _____ ______ _________ ______ ____.
__________ ________ __________ ________ _____ _________ _______ _______ _______ __________ ________.
_________ __________ _____ ________ ____.
______ _______ ________ ______ ________ ________.
_____ ______ __________ _______ ______ ________ ___ _________ ____ ______.
_______ __________ ______ _______ __________ ______ _______ _________ _______.
________ ______ ______ _____ ________ ________ __________ ____ ________.
__________ ________ __________ _____ ____ _______ _____.
_________ __________ _________ ______ ______ ____ ___ _________ ___ _______.
________ _______ ________ ______ ____ _________ ____.
________ __________ _______ ______ ______ ___.
____ ____ ___ ____ ____ _____.
______ ________ ____ ________ ___ __________ ___ ______ ____ _______.
__________ _____ ___ __________ ____ _________ _________ ____ ________ _________ _________ ____.
______ _____ ____ ____ ____ ________ ________ _____.
___ __________ __________ ____ _______ _______.
____ ______ ____ ________ _____ _______ __________ _____ _____.
______ ___ _________ _______ _____.
________ ____ ___ _________ ___ ______ ______ ____ _________ ____.
_____ ___ ____ _________ __________ ______ ________.
______ ___ __________ ________ _________ _______ _________ ___.
____ _____ _________ _____ _______ ________ ____ __________ _______ ___ _____ __________.
_________ _______ ______ _____ ____ _____ ______ _____ _____ _________ ____.
___ __________ _______ ____ ____ __________ ___ ________ ____ _________ __________.
________ ____ _________ _____ _____.
___ _________ ________ ____ _____ ________ _______.
___ ________ ___ ______ _____ __________ ___ ________ _____.
________ ___ _______ ____ _____ _____.
____ _______ ________ ______ _________ ___ _________ __________ __________ ________.
_______ ________ ____ ________ ___ ___ ______ ______ ________ _______ _________.
_______ _____ ___ __________ _________.
_____ ______ ______ ____ ___ ______ ________ ____ ___ __________.
______ _____ _____ ________ ______ ________ ____ ________ ________.
___ ________ ______ ______ _________ _________ _______ _________ _____ __________ _________.
________ _________ ________ __________ ___ ___ ____.
_______ ___ _______ ____ ___ ____ ___ _______ ____ ___ _____ __________.
______ ______ _____ ___ ___ _____ __________ ____ ______ _______.
________ _______ ______ __________ __________ ______ ________ ____ _________ ____ __________.
_______ __________ ___ ____ _______ ______ ______ _______ __________ __________ _________.
________ ___ ____ _______ _________ ___ _____ _______.
____ ___ __________ __________ ___ ___ _____.
_______ _______ ___ _________ _________ ___ _______ _____ ________ __________ ____ ________.
______ ____ _________ _______ ________.
________ ___ _______ _____ _____ ____ _____ ____.
__________ ________ _________ ____ ______ _______ ___ _________ ___.
________ ____ ___ _________ ___ ______ _________ ____ __________.
_____ _________ ________ _____ ________ _________ _______ _______ ___.
_____ ________ ________ ____ ______ _____ ___ ________.
_________ ______ __________ ____ _______ __________ ___ _____ _______.
________ _____ __________ ____ __________ _______ __________ ____ ________ __________ ______ ___.
______ ________ ___ __________ ________ ____ _______ ______ _____ ______ __________.
_______ ____ ______ _______ ______ _____ _________ __________ __________ _____ ___ _________.
_____ _________ ________ _________ _______ ________ __________ _____ _____.
_____ ______ ____ _____ __________ _____ ________.
_____ __________ ________ ____ ________ _________ ___ __________.
_________ ______ _________ ________ _______ ________ ____ _______ ____.
__________ _______ ____ ___ _______ __________ ______ __________ ________ __________.
_______ ________ _________ ______ ____ __________.
___ _____ _______ _________ ______ __________ ________ _____.
______ ___ __________ _____ ____ ______ ___ ________ ___ _______.
_____ ____ __________ ____ ___.
_____ ______ _____ ___ ________ ______.
________ _______ ___ ____ ______.
______ _______ __________ _____ _____ _______ __________ ___ __________ _______ __________.
_______ ______ ______ _________ ___ _____ __________ ______ _______ ___.
________ ____ __________ ___ _____ _____.
_____ _______ ______ _______ ______.
_______ _______ _______ _______ _________ __________ __________ _________ _______.
________ ______ ________ __________ ___ _______ ________ __________.
______ _________ ______ _________ _______.
__________ _____ _________ _________ ______ _______ _______ ______ _____ __________ ___.
______ ________ ___ _________ ______ __________.
_________ ___ __________ _____ _______ _____ ___ __________ __________ ____.
______ ________ ______ ____ _________ ______ ___ ______ _________ _________ __________.
___ ____ __________ ______ ______ _______ ___ _____ __________ _____.
______ __________ _______ __________ ____ __________.
Get Full Answer on WhatsApp