Question

Rewrite the linear system as a matrix equation \(\mathbf{y}^{\prime}=A \mathbf{y}\), and compute the eigenvalues of the matrix \(A\). $$ \begin{aligned} &y_{1}^{\prime}=3 y_{1}+2 y_{2} \\ &y_{2}^{\prime}=-4 y_{1}-3 y_{2} \end{aligned} $$

Short answer

Based on the given linear system of equations, rewrite it as a matrix equation and find the eigenvalues of the matrix. The linear system of equations is: $$ \begin{aligned} &y_{1}^{\prime} = 3 y_{1} + 2 y_{2} \\ &y_{2}^{\prime} = -4 y_{1} - 3 y_{2} \end{aligned} $$ Rewritten as a matrix equation: $$ \mathbf{y}^{\prime} = A \mathbf{y} = \begin{bmatrix} 3 & 2 \\ -4 & -3 \end{bmatrix} \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} $$ The eigenvalues of the matrix \(A\) are \(5\) and \(-5\).

Step-by-step solution

Rewrite the linear system as a matrix equation: \(\mathbf{y}^{\prime} = A \mathbf{y}\)

To rewrite the given linear system as a matrix equation, we can extract the coefficients of \(y_1\) and \(y_2\) from the equations and use them as the elements of matrix \(A\). For the given linear system: $$ \begin{aligned} &y_{1}^{\prime} = 3 y_{1} + 2 y_{2} \\ &y_{2}^{\prime} = -4 y_{1} - 3 y_{2} \end{aligned} $$ We can identify the matrix \(A\) as: $$ A = \begin{bmatrix} 3 & 2 \\ -4 & -3 \end{bmatrix} $$ So, the matrix equation will be: $$ \mathbf{y}^{\prime} = A \mathbf{y} = \begin{bmatrix} 3 & 2 \\ -4 & -3 \end{bmatrix} \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} $$

Compute the eigenvalues of matrix \(A\)

To find the eigenvalues of matrix \(A\), we need to solve the characteristic equation, which is given by: $$ \det(A - \lambda I) = 0 $$ Where \(I\) is the identity matrix, and \(\lambda\) denotes the eigenvalue. First, calculate the matrix \(A - \lambda I\): $$ A - \lambda I = \begin{bmatrix} 3 - \lambda & 2 \\ -4 & -3 - \lambda \end{bmatrix} $$ Now, compute the determinant: $$ \det(A - \lambda I) = (3-\lambda)(-3-\lambda) - (-4)(2) = \lambda^2 - 25 $$ Set the determinant to zero and solve for \(\lambda\): $$ \lambda^2 - 25 = 0 $$ The solutions to this quadratic equation are: $$ \lambda_1 = 5, \quad \lambda_2 = -5 $$ So, the eigenvalues of matrix \(A\) are \(5\) and \(-5\).

Key concepts

Linear Systems

Linear systems are sets of equations that involve linear combinations of variables. In our exercise, the linear system consists of two equations:

• \( y_{1}^{\prime} = 3y_1 + 2y_2 \)
• \( y_{2}^{\prime} = -4y_1 - 3y_2 \)

Here, \( y_1 \) and \( y_2 \) are variables, and their derivatives \( y_1^{\prime} \) and \( y_2^{\prime} \) represent changes concerning time or another variable. Linear systems like these can be transformed into matrix equations, making them easier to analyze and solve. The main goal is to extract coefficients from these equations and organize them in a matrix form, simplifying further computations.

Matrix Equations

Matrix equations provide a compact way to represent linear systems through matrices and vectors. This simplifies solving systems of equations, especially when dealing with large or complex systems. In our example, the linear system is rewritten as a matrix equation \( \mathbf{y}^{\prime} = A \mathbf{y} \). The matrix \( A \) captures the coefficients of the original equations:

• \( A = \begin{bmatrix} 3 & 2 \ -4 & -3 \end{bmatrix} \)

This matrix acts on the vector \( \mathbf{y} = \begin{bmatrix} y_1 \ y_2 \end{bmatrix} \) representing the variables. By organizing the equations in this matrix format, it becomes straightforward to apply linear algebra techniques such as finding eigenvalues, which help understand the behavior of the system.

Differential Equations

Differential equations involve equations with derivatives and are crucial for modeling systems where quantities change over time. In our scenario, \( y_1^{\prime} \) and \( y_2^{\prime} \) signify the rates of change of \( y_1 \) and \( y_2 \), respectively. These first-order linear differential equations describe how two interrelated quantities can evolve over time based on their current states. By converting these equations into a matrix form, we can easily apply mathematical techniques such as eigenvalue analysis, which helps in predicting the system's long-term behavior or stability.

Determinants

Determinants are a fundamental concept in matrix algebra. They provide a scalar value that indicates certain properties of a matrix, such as invertibility and eigenvalues. For a matrix \( A \), the determinant helps determine the solutions of the equation \( \det(A - \lambda I) = 0 \), where \( I \) is the identity matrix and \( \lambda \) is an eigenvalue.Here, the determinant of the matrix \( A - \lambda I \) is computed as:

• \( \det(A - \lambda I) = (3-\lambda)(-3-\lambda) - (-4)(2) = \lambda^2 - 25 \)

Setting this determinant to zero allows us to solve for \( \lambda \), revealing the eigenvalues \( 5 \) and \( -5 \). These eigenvalues are crucial for understanding the system's dynamics, indicating growth rates or oscillations in the solutions.