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sample - ChBE 521 Sample Problems for Exam 2 1 Find the...

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ChBE 521 Sample Problems for Exam 2 1. Find the eigenvalues and eigenvectors, and the diagonalizing matrix, for A = 1 0 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 = 0 1 0 0 = 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 ). 8. If A 2 = - I , what are the eigenvalues of A ? If A is a real n by n matrix show that n must be even and give an example.
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