SIT292 LINEAR ALGEBRA 2013 Assignment 3 Due: 5 p.m. 23 September, 2013 1. For the matrices A = 24 1 2 3 0 ??1 2 0 0 3 35; B = 24 ??1 ??2 3 0 1 ??5 0 0 3 35 (a) nd the eigenvalues and eigenvectors (b) determine a matrix P so that P??1AP = B( 10 + 10 = 20 marks) 2. For the following matrixA = 2664 9 ??1 8 6 ??1 5 ??5 1 ??4 3775 (a) nd the eigenvalues (b) for each eigenvalue determine the eigenvector(s) (c) determine a matrix P so that B = P??1AP is in triangular form, and verify that the determinant of B agrees with what you used in (a) ( 5 + 10 + 5 = 20 marks)
3. For the following matrix A = 24 1 1 0 2 1 0 1 1 0 1 1 1 35 (a) determine the row-rank (b) nd 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. ( 6 + 2 + 6 = 14 marks) 4. For the following matrix 2664 6 3 3 3 6 3 3 3 6 3775 (a) nd the eigenvalues (b) nd 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 + 8 + 5 = 18 marks) 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. ( 8 marks)