Question

Find the eigenvalues and corresponding vectors of the matrix A=[ 0 0 1 / 0 1 0 / -1 0 0] 1. Calculate the eigenvalues λ₁, λ₂, and λ₃ of matrix A. 2. For each eigenvalue λi, find the corresponding eigenvectors and show the working for each.

Answer 1

Eigenvalues and corresponding eigenvectors of a matrix are determined by solving the characteristic equation and the linear equation (A - λI)v = 0 respectively; this entails finding the determinant of a modified matrix and solving a system of equations.

Step-by-step explanation:

Finding Eigenvalues and Eigenvectors

To find the eigenvalues λ1, λ2, and λ3 of matrix A = [0 0 1; 0 1 0; -1 0 0], we solve the characteristic equation det(A - λI) = 0:

1. Subtract λ from the diagonal elements of A.
2. Calculate the determinant of the resulting matrix.
3. Solve for λ, which gives us the eigenvalues.

Once the eigenvalues are found, we find the corresponding eigenvectors by solving the system of linear equations (A - λI)v = 0, where v is the eigenvector associated with eigenvalue λ.

For each eigenvalue λi, we substitute λi into the equation and find the solutions for the vector v which satisfy the system. This involves reducing the system to row-echelon form and finding a basis for the null space.