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Unformatted text preview: THE UNIVERSITY OF SYDNEY PURE MATHEMATICS Linear Mathematics 2012 Practice session 5 1. Consider the following two matrices: A = parenleftBig 0 0- 2 1 2 1 1 0 3 parenrightBig and B = parenleftBig 3 1 2 4 parenrightBig . For each of these matrices: a ) Find all the eigenvalues for the matrix and, for each eigenvalue, find a basis for the corre- sponding eigenspace. b ) If possible, find a basis of R 3 for A , and of R 2 for B , consisting of eigenvectors. c ) Write down the square matrix P whose columns are the basis vectors you found in part (b). (Such a matrix is invertible. Why?) Write down the diagonal matrix D whose diagonal entries are the eigenvalues of the matrix, in the same order as the corresponding columns of P . d ) Check by doing the multiplications that AP = PD . (This confirms that A is diagonalis- able; that is, P- 1 AP = D .) 2. Let A = parenleftBig 2 0- 2 0 3 0 0 0 3 parenrightBig . Show that A is diagonalizable and hence find A n , for n ≥ ....
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This note was uploaded on 02/06/2012 for the course MATH 2061 taught by Professor Notsure during the Three '09 term at University of Sydney.
- Three '09