Question

Consider the linear system
y'=[[ -3 -2; 6 4 ]] y
a. Find the eigenvalues

Answer 1

The eigenvalues of the matrix A = [ [-3 -2]; [6 4] ] are found by factoring the characteristic equation λ^2 - λ - 12 = 0, resulting in two eigenvalues: 3 and -4.

Step-by-step explanation:

To find the eigenvalues of the given linear system, we need to consider the matrix A = [ [-3 -2]; [6 4] ]. The eigenvalues λ are found by solving the characteristic equation, det(A - λI) = 0, where I is the identity matrix.

The characteristic equation in this case is \((-3 - λ)(4 - λ) - (-2)(6) = λ^2 - λ - 12 = 0.

Factoring the quadratic equation, we get two eigenvalues: λ1 = 3 and λ2 = -4. These are the eigenvalues of the matrix A, which correspond to the rates at which solutions of the system grow or decay.