# Eigenvalues of coefficient matrix problem

• DrunkApple
In summary, the conversation discusses determining the general solution of a system of homogeneous differential equations. This involves finding the Eigenvalues and corresponding eigenvectors of the coefficient matrix, and using them to find expressions for x_{1}(t) and x_{2}(t). The quadratic equation can be used to find the Eigenvalues.
DrunkApple

## Homework Statement

Determine the general solution of the system of homogeneous differential equations.
The system of homogeneous differential equations is:
X'$_{1}$(t) = 141x$_{1}$(t) - 44x$_{2}$(t)
X'$_{2}$(t) = 468x$_{1}$(t) - 146$_{2}$(t)

What is Eigenvalues of Coefficient Matrix?
What is corresponding eigenvectors?
What is expression for x$_{1}$(t)?
What is expression for x$_{2}$(t)

## The Attempt at a Solution

What is Eigenvalues of Coefficient Matrix?
|144 - 44 | = A
|146 -146|
det|144-λ -44 | = P(λ)
|468 -146-λ|
= (144-λ)(-146-λ) - (-44)(468)
= -21024 -144λ +146λ + λ$^{2}$ + 20592
= λ$^{2}$ +2λ -432
Do I use quadratic equation to find λ?

yes of course, then you find the eigenvectors, then you find your solution to the DE using these eigenvectors

## 1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used to understand the behavior of a linear transformation or a matrix. An eigenvector is a non-zero vector that, when multiplied by a matrix, results in a scalar multiple of itself. The corresponding scalar value is called an eigenvalue.

## 2. How are eigenvalues and eigenvectors related to the coefficient matrix problem?

In the coefficient matrix problem, the matrix represents a set of linear equations. The eigenvalues and eigenvectors of this matrix provide important information about the solutions to these equations. Specifically, the eigenvalues determine the stability of the system and the eigenvectors provide the direction of the system's behavior.

## 3. What is the significance of finding eigenvalues in the coefficient matrix problem?

Finding eigenvalues in the coefficient matrix problem can help us understand the behavior of the system represented by the matrix. It can provide insights into the stability of the system and how it will evolve over time. Eigenvalues also play a crucial role in solving differential equations and other important mathematical problems.

## 4. How do we calculate eigenvalues of a coefficient matrix?

To calculate the eigenvalues of a coefficient matrix, we need to find the values of lambda that satisfy the characteristic equation det(A-λI) = 0, where A is the coefficient matrix and I is the identity matrix. The resulting values of lambda are the eigenvalues of the matrix.

## 5. Can a coefficient matrix have complex eigenvalues?

Yes, a coefficient matrix can have complex eigenvalues. This is because the characteristic equation can result in complex roots, which correspond to complex eigenvalues. These complex eigenvalues can provide important information about the behavior of the system, such as oscillations or spiraling behavior.

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