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Unformatted text preview: ChBE 521 Sample Problems for Exam 2 1. Find the eigenvalues and eigenvectors, and the diagonalizing matrix, for A = " 1 2 3 # and B = " 7 2- 15- 4 # 2. If A has eigenvalues 0 and 1, corresponding to eigenvectors " 1 2 # and " 2- 1 # and how can you tell that the matrix is symmetric? 3. In the previous problem, what are the eigenvalues and eigenvectors for A 2 ? What is the relation of A 2 to A . 4. If A and B are diagonalizable, then is AB ? 5. By trying to solve " a b c d #" a b c d # = " 1 # = A Show that A has no square root. Change the diagonal entiries of A to 4 and try to find a square root. 6. True or false: (a) Every invertible matrix can be diagonalized. (b) Every diagonalizable matrix can be inverted. (c) Changing the rows of a 2x2 matrix changes the sign of the eigenvalues. (d) If eigenvectors x and y correspond to distinct eigenvalues, then x T y = 0. 7. If K is a skew-symmetric matrix ( K =- K T ), show that Q = ( I- K )( K + K )- 1 is an orthogonal matrix ( Q T Q = I )....
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This note was uploaded on 11/30/2010 for the course CHBE ChaBE521 taught by Professor Chrisrao during the Spring '10 term at University of Illinois, Urbana Champaign.
- Spring '10