Eigenvalues and Eigenvectors

Part 1: Calculation of Eigenvalues and Eigenvectors

1. The matrices A, B, C, and the 3x3 identity matrix I₃ are defined in your worksheet. Calculate the characteristic polynomial of A by taking the determinant of A - lambda I₃ Factor the polynomial.
2. Solve the linear system (A - I₃) v = 0 by finding the reduced row echelon form of A - I₃. Explain why this enables you to write down an eigenvector corresponding to the eigenvalue 1.
3. Determine an eigenvector v corresponding to the eigenvalue 1. Check that this is correct by calculating Av and v. Describe all eigenvectors of A that correspond to the eigenvalue 1.
4. Repeat the process in Steps 2 and 3 for the other eigenvalues of A, and record the results.
5. Now find the eigenvalues of B, and for each eigenvalue describe all the corresponding eigenvectors. Repeat this process for C. How do the calculations for A, B, and C differ?
6. Now check your eigenvalue and eigenvector calculations using the appropriate commands in your computer algebra system. Explain any differences in the output of the commands for matrices A, B, and C.