Question

"for each of the following matrices compute the eigenvalues and eigenvectors and indicate the type of the system (saddle point, node (or sink), source, center, spiral source, spiral sink).a. A= ([-9 8] [-3 1])b. A= ([6 -2] [4 2])c. A= ([-4 5] [-5 4])"

Answer 1

To compute the eigenvalues and eigenvectors of the given matrices, we can start by finding the eigenvalues. Then, we can use the eigenvalues to find the eigenvectors. Finally, we can determine the type of system based on the eigenvalues.

Step-by-step explanation:

To compute the eigenvalues and eigenvectors of each given matrix, we can start by finding the eigenvalues. The eigenvalues are the values λ that satisfy the equation det(A - λI) = 0, where A is the matrix and I is the identity matrix. For matrix A:

A = [-9 8] [-3 1]

Using the formula for the determinant of a 2x2 matrix, we have:

det(A - λI) = (-9 - λ)(1 - λ) - (-3)(8) = λ^2 - (-10)λ + 15 = λ^2 + 10λ + 15 = 0

Factoring the quadratic equation, we find: (λ + 3)(λ + 5) = 0

So the eigenvalues for matrix A are λ = -3 and λ = -5.

To find the eigenvectors, we substitute each eigenvalue back into the equation (A - λI)x = 0, and solve for x. For the eigenvalue λ = -3:

(A - (-3)I)x = 0

[-6 8] [-3 4]x = 0

From here, we can find the eigenvectors by solving the system of equations -6x + 8y = 0 and -3x + 4y = 0. The solutions are x = 2y, where y is any non-zero real number. So, the eigenvector corresponding to λ = -3 is [2y y], where y is any non-zero real number.

To determine the type of system, we can look at the eigenvalues. If both eigenvalues are negative, the system is a sink or stable node. If both eigenvalues are positive, the system is a source or unstable node. If one eigenvalue is positive and one is negative, the system is a saddle point. For matrix A, since the eigenvalues are λ = -3 and λ = -5 (both negative), the system is a sink or stable node.

Answer 2

To find eigenvalues and eigenvectors, solve the characteristic equation and system equation respectively. Types of systems are determined by the sign and nature of the eigenvalues. Saddle points, nodes, sources, sinks, centers, and spirals are identified through these values.

Step-by-step explanation:

To compute the eigenvalues of a matrix, we solve the characteristic equation det(A - λI) = 0, where A is our matrix, λ represents the eigenvalues, and I is the identity matrix. For eigenvectors, we solve the equation (A - λI)x = 0 for x, where x is the eigenvector corresponding to the eigenvalue λ.

b. Matrix A = [[6 -2] [4 2]] will have its eigenvalues calculated similarly, followed by eigenvectors.

c. Matrix A = [[-4 5] [-5 4]] will have complex eigenvalues leading to either a spiral source or spiral sink, which is identified by the real part of the eigenvalues.

The type of system is determined by the eigenvalues' real parts: a saddle point has eigenvalues of opposite signs, a node has real, same-sign eigenvalues, a source has positive real parts, a sink has negative real parts, centers have purely imaginary eigenvalues, a spiral source has complex eigenvalues with positive real parts, and a spiral sink has complex eigenvalues with negative real parts.