Question

Find the eigenvalues and eigenvectors of the given matrix.

Answer 1

To find the eigenvalues and eigenvectors of a matrix, calculate the characteristic equation by setting the determinant of the matrix subtracted by λ times identity matrix to zero. Solve for λ to get eigenvalues, and substitute each into the equation (A-λI)x = 0 to find the corresponding eigenvectors.

Step-by-step explanation:

Finding Eigenvalues and Eigenvectors To find the eigenvalues and eigenvectors of a given matrix, you need to perform several steps. First, you need to find the eigenvalues by solving the characteristic equation, which is derived from subtracting λ times the identity matrix from the original matrix and setting the determinant of the resulting matrix to zero. Once the eigenvalues are found, eigenvectors are obtained by substituting each eigenvalue into the equation (A-λI)x = 0 and solving for the vector x.

Here are the steps in more detail:

1. Subtract λ (an eigenvalue) times the identity matrix from the original matrix to get the matrix (A-λI).
2. Set the determinant of (A-λI) to zero and solve for λ to get the eigenvalues.
3. For each eigenvalue λ, substitute it back into (A-λI)x = 0 and solve for the eigenvector x.

It's important to apply these steps carefully to ensure that the eigenvalues and eigenvectors are found correctly.