Question

In exercises 5 and 6, the matrix (A) is factored in the form (PDP^-1). According to the diagonalization theorem, how can you find the eigenvalues of matrix (A)?

a) Solve the characteristic equation det((A - λ I)) = 0
b) Take the determinant of matrix (P)
c) Find the inverse of matrix (D)
d) Use the trace of matrix (A)

Answer 1

To find the eigenvalues of matrix A, solve the characteristic equation det((A - λ I)) = 0.

Step-by-step explanation:

To find the eigenvalues of matrix A, you need to solve the characteristic equation det((A - λ I)) = 0, where λ is an eigenvalue.

Multiply matrix A by the identity matrix (I) and subtract λI from matrix A.

Calculate the determinant of (A - λ I) and set it equal to zero.

Answer 2

a) Solve the characteristic equation det((A - λ I)) = 0

Step-by-step explanation:

The correct method for finding the eigenvalues of matrix \(A\) according to the diagonalization theorem is option (a): solving the characteristic equation det((A - λ I)) = 0. The characteristic equation is obtained by setting the determinant of the matrix (A - λ I) equal to zero, where λ is the eigenvalue and I is the identity matrix. The solutions to this equation are the eigenvalues of matrix A.

To elaborate, let A be an n × n matrix. The characteristic equation is given by det((A - λ I)) = 0, where det denotes the determinant, λ represents the eigenvalue, A is the matrix, and I is the identity matrix. Solving this equation yields the eigenvalues of A. Each solution corresponds to an eigenvalue λ, and once these eigenvalues are determined, they can be used to further analyze the properties and behavior of the matrix.

In summary, the eigenvalues of matrix A are found by solving the characteristic equation det((A - λ I)) = 0. This equation encapsulates the relationship between the matrix A and its eigenvalues, providing a fundamental tool in linear algebra for understanding the intrinsic properties of matrices.

Full Question

According to the diagonalization theorem, how can you find the eigenvalues of matrix \(A\)?

a) Solve the characteristic equation det((A - λ I)) = 0

b) Take the determinant of matrix \(P\)

c) Find the inverse of matrix \(D\)

d) Use the trace of matrix \(A\)