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The matrix method is a powerful tool to solve systems of differential equations. By representing the system in matrix form, mathematical operations become more structured and manageable. Suppose we have a system of differential equations involving several variables, like in our exercise. We begin by expressing this system as a matrix equation.
Imagine we have variables \( x, y, \) and \( z \) such that we write \( \mathbf{X} = \begin{pmatrix} x & y & z \end{pmatrix}^T \) and a corresponding matrix \( A \). The system can be simplified to a single matrix equation: \( \mathbf{X}' = A \mathbf{X} \). Here, \( \mathbf{X}' \) denotes the vector of first derivatives of \( x, y, z \), and \( A \) is a square matrix comprising the coefficients from the original system.

Eigenvalues are crucial in solving the matrix equation for differential systems. To find them, we first compute the characteristic polynomial: \(|A - \lambda I| = 0 \). This involves subtracting \( \lambda I \) from the matrix \( A \), where \( I \) is the identity matrix of the same size.
The resulting expression is a polynomial equation in \( \lambda \). It can be solved by expanding the determinant, simplifying the terms, and finding the values of \( \lambda \) that satisfy the equation. These values are the eigenvalues of \( A \), and each represents a 'scale' factor in the system dynamics.

Once we have the eigenvalues, we move on to finding the eigenvectors. An eigenvector corresponds to each eigenvalue \( \lambda \) and gives further insight into the system. To find an eigenvector \( \mathbf{v} \) associated with \( \lambda \), we solve the equation \( (A - \lambda I)\mathbf{v} = 0 \).
This means subtracting \( \lambda \) times the identity matrix from \( A \) and solving for \( \mathbf{v} \). The solution usually involves setting up a system of linear equations. The non-trivial solution (vector that isn't just zero) to this system is the eigenvector. Eigenvectors show the directions in which the transformation defined by \( A \) scales the space.

The general solution of the system of differential equations can be constructed using the eigenvalues and eigenvectors. Given a matrix \( A \) with eigenvalues \( \lambda_1, \lambda_2, \lambda_3 \) and corresponding eigenvectors \( \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3 \), the solution can be written as: \( \mathbf{X}(t) = c_1 e^{\lambda_1 t} \mathbf{v}_1 + c_2 e^{\lambda_2 t} \mathbf{v}_2 + c_3 e^{\lambda_3 t} \mathbf{v}_3 \). Here, \( c_1, c_2, c_3 \) are constants determined by initial conditions.
Each term \( e^{\lambda t} \mathbf{v} \) represents a component of the solution that evolves over time, influenced by the corresponding eigenvalue's growth or decay rate (depending on if \( \lambda \) is positive or negative).

A system of differential equations consists of multiple interrelated differential equations. Instead of a single equation with a single function, we deal with multiple functions and their derivatives.
These systems often model complex phenomena where several quantities interact, like in our exercise with functions \( x(t), y(t) \), and \( z(t) \). Solutions to such systems provide a comprehensive description of how the quantities evolve over time, given their interdependencies.