Question

In Exercises \(1-16\) find the general solution. $$ \mathbf{y}^{\prime}=\left[\begin{array}{rrr} 1 & 1 & 2 \\ 1 & 0 & -1 \\ -1 & -2 & -1 \end{array}\right] \mathbf{y} $$

Short answer

To summarize, we found the eigenvalues of the given matrix as \(\lambda = 0, -1, -2\) and their corresponding eigenvectors as \(\mathbf{v}_1 = \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix},\) and \(\mathbf{v}_3=\begin{bmatrix} 1/2 \\ -1 \\ 1 \end{bmatrix}\). The general solution of the given system of differential equations is given by: $$ \mathbf{y}(t) = c_1 e^{0t}\begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix} + c_2 e^{-t}\begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix} + c_3 e^{-2t}\begin{bmatrix} 1/2 \\ -1 \\ 1 \end{bmatrix}, $$ where \(c_1, c_2,\) and \(c_3\) are arbitrary constants.

Step-by-step solution

Find the eigenvalues of the matrix

In order to find the eigenvalues, we first need to compute the determinant of the matrix \((A-\lambda I)\), where \(A\) is the given matrix, \(\lambda\) is the eigenvalue, and \(I\) is the identity matrix. We have: $$ \det\left(A-\lambda I\right)=\det\begin{bmatrix} 1-\lambda & 1 & 2 \\ 1 & -\lambda & -1 \\ -1 & -2 & -1-\lambda \end{bmatrix}. $$ Computing the determinant, we get: $$ (1-\lambda)[(-\lambda)(-1-\lambda)+(-1)(-2)] - 1[(-1-\lambda) - (-2)(-1)] + 2[(-1)(-1)-(-\lambda)] = 0. $$ Simplifying the expression, we obtain: $$ (-\lambda^3 - 2\lambda^2 - 3\lambda) = 0. $$ The solutions to this equation are the eigenvalues of the given matrix, which are \(\lambda=0, -1,-2\).

Find the eigenvectors for each eigenvalue

Now, we compute the eigenvectors corresponding to each eigenvalue by solving the equation \((A-\lambda I)\mathbf{v}=0\), where \(\mathbf{v}\) is an eigenvector. For \(\lambda=0\), we have: $$ \begin{bmatrix} 1 & 1 & 2 \\ 1 & 0 & -1 \\ -1 & -2 & -1 \end{bmatrix}\begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}=0. $$ Solving this system of equations, we get \(\mathbf{v}_1=\begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}\) as an eigenvector. For \(\lambda=-1\), we have: $$ \begin{bmatrix} 2 & 1 & 2 \\ 1 & 1 & -1 \\ -1 & -2 & 0 \end{bmatrix}\begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}=0. $$ Solving this system of equations, we get \(\mathbf{v}_2=\begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix}\) as an eigenvector. For \(\lambda=-2\), we have: $$ \begin{bmatrix} 3 & 1 & 2 \\ 1 & 2 & -1 \\ -1 & -2 & 1 \end{bmatrix}\begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}=0. $$ Solving this system of equations, we get \(\mathbf{v}_3=\begin{bmatrix} 1/2 \\ -1 \\ 1 \end{bmatrix}\) as an eigenvector.

Write the general solution

Now that we have the eigenvalues and their corresponding eigenvectors, we can express the general solution of the given system of differential equations as follows: $$ \mathbf{y}(t) = c_1 e^{0t}\begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix} + c_2 e^{-t}\begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix} + c_3 e^{-2t}\begin{bmatrix} 1/2 \\ -1 \\ 1 \end{bmatrix}, $$ where \(c_1,c_2,\) and \(c_3\) are arbitrary constants. This is the general solution of the given system of differential equations.

Key concepts

Determinant Calculation

Determinant calculation is a crucial operation in linear algebra that is used to perform a variety of tasks, including solving systems of linear equations, evaluating if a matrix is invertible, and finding eigenvalues for matrices. The determinant of a square matrix, often represented by \( \det(A) \), can be thought of as a scalar value that summarizes certain properties of the matrix.

When solving differential equations that involve matrices, like in our original exercise, you must often find the eigenvalues of a given matrix. To do this, you'll need to compute the determinant of the matrix after subtracting \( \lambda \) times the identity matrix from it. This operation gives us a polynomial equation known as the characteristic equation, which we then solve for \( \lambda \) to find the eigenvalues. This process relies heavily on adequate determinant calculation abilities, which allow you to transform a complex matrix into a simple polynomial that reveals the matrix's eigenvalues.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in the study of linear algebra and differential equations. An eigenvalue is a special scalar that is associated with a linear transformation represented by a matrix, and an eigenvector is a vector that only scales under the linear transformation, rather than also being rotated.

When we talk about finding eigenvalues, we refer to solving the characteristic polynomial obtained from the determinant calculation. After determining the eigenvalues, we proceed to find the corresponding eigenvectors. This is done by substituting each eigenvalue back into the equation \( (A-\lambda I)\mathbf{v}=0 \), and solving for the vector \( \mathbf{v} \).

In the context of systems of differential equations, eigenvalues can inform us about the behavior of the system over time, such as stability and oscillation, while eigenvectors help to determine the direction of the system's trajectories.

Systems of Differential Equations

Systems of differential equations are collections of two or more equations involving derivatives of several unknown functions. These systems can depict numerous real-world processes, from physics to economics. In the case of linear systems, they can often be written in matrix form as \( \mathbf{y}^\prime = A\mathbf{y} \), where \( A \) is a matrix of coefficients, \( \mathbf{y} \) is the vector of unknown functions, and \( \mathbf{y}^\prime \) is the vector of their derivatives.

To solve such systems, the eigenvalue-eigenvector approach is particularly useful, especially for homogeneous linear systems with constant coefficients, like the one in our example. By finding the eigenvalues and eigenvectors, you can construct the general solution to the system with exponential functions whose exponents are the eigenvalues, and the multipliers are the eigenvectors, combined linearly with arbitrary constants. This method creates a powerful toolkit for solving complex dynamic systems that are prevalent in many disciplines.