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Unformatted text preview: Department of Mathematical Sciences University of Delaware Prof. T. Angell November 11, 2009 Mathematics 351 Exercise Sheet 7 Exercise 31: Compute the inverse of the following using elementary row operations 12 1 cos (θ) 0 − sin (θ) . 3. 0 1 0 (a) A = 2 1 (b) B = 0 3 −1 sin (θ) 0 cos (θ) Exercise 32: Find the LU factorization of the matrix 23 A=4 1 34 1 4 6 Exercise 33: Suppose that Gaussian elimination gives the solution of a linear system Ax = c as x = xo + α1 x1 + α2 x2 , where A is a 6 × 6 matrix and α1 and α2 are arbitrary. Is the matrix A invertible? Explain. Exercise 34: We say that a matrix N is nilpotent provided that, for some integer p, N p = 0. (a) Show that every nilpotent matrix is singular. (b) Show that if N is nilpotent (I − N )−1 = I + N + N 2 + · · · + N p−1 . (c) Write the matrix 1 0 A= 0 0 5 1 0 0 1 2 1 0 0 7 . 3 1 in the form I + N where N is nilpotent and ﬁnd the inverse of A. Exercise 35: An n × n matrix Q is said to be orthogonal provided Q￿ Q = I . (a) Show that if Q is orthogonal so is Q−1 . (b) Show that if Q is orthogonal its rows form an orthonormal set of vectors. In other words, show that the dot product of a row with itself is 1 and the dot product of two diﬀerent rows is 0. (c) What possible values can the determinant of an orthogonal matrix have? Explain. Due Date: Wed. October 18, in class ...
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This note was uploaded on 12/02/2009 for the course MATH 352 taught by Professor Staff during the Spring '08 term at University of Delaware.

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