1. For the matrices
![670_Find the eigenvalues and eigenvectors.png](https://secure.tutorsglobe.com/CMSImages/670_Find%20the%20eigenvalues%20and%20eigenvectors.png)
(a) find the eigenvalues and eigenvectors
(b) determine a matrix P so that P-1 AP = B
2. For the following matrix
![851_Find the eigenvalues and eigenvectors1.png](https://secure.tutorsglobe.com/CMSImages/851_Find%20the%20eigenvalues%20and%20eigenvectors1.png)
(a) find the eigenvalues
(b) for each eigenvalue determine the eigenvector(s)
(c) determine a matrix P so that B = P-1 AP is in triangular form, and verify that the determinant of B agrees with what you used in (a)
3. For the following matrix
![1783_Find the eigenvalues and eigenvectors2.png](https://secure.tutorsglobe.com/CMSImages/1783_Find%20the%20eigenvalues%20and%20eigenvectors2.png)
(a) determine the row-rank
(b) find a set of generators for the row space of A
(c) show that any element of the row space of A can be written as a linear combination of your generators.
4. For the following matrix
![614_Find the eigenvalues and eigenvectors3.png](https://secure.tutorsglobe.com/CMSImages/614_Find%20the%20eigenvalues%20and%20eigenvectors3.png)
(a) find the eigenvalues
(b) find the eigenvectors corresponding to these eigenvalues
(c) starting with the eigenvectors you found in (a) construct a set of orthonormal vectors (use the Gram-Schmidt procedure).
5. Check whether the set of ordered triples f(2, 0, 2), ( 1, 2, 1), (1, 1, 1)g forms a basis for R3. If so, starting with this basis use the Gram-
Schmidt procedure to construct an orthonormal basis for R3.