hw4 - A = ± α 4 1 α ² B = 2 α-3-1 5 1 3 α For what...

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Homework 4 Math 471, Fall 2007 Assigned: Friday, September 28, 2007 Due: Friday, October 5, 2007 Include a cover page Clearly label all plots using title , xlabel , ylabel , legend Use the subplot command to compare multiple plots Include printouts of all Matlab code, labeled with your name, date, section, etc. (1) (Laguerre’s Method, P. 135 #2d) Apply Laguerre’s method to the polynomial p ( x ) = x 4 - 2 x 3 - 5 x 2 +12 x - 5 with a starting approximation of x 0 = 0 and a convergence tolerance of 5 × 10 - 11 . (2) (Matrix Operations, P. 148 #1-9) Compute the indicated matrices given A = 1 - 1 3 2 0 5 , B = 2 1 0 - 3 - 1 5 1 3 4 , C = 4 2 3 - 1 2 - 4 and D = 1 - 1 4 0 2 - 2 0 0 3 . 1 . (a) 2 A + C T , (b) C - 3 B ; 2 . (a) AB , (b) AD ; 3 . (a) CA , (b) AC ; 4 . (a) BD , (b) DB ; 5 . (a) BC , (b) CB ; 6 . (a) 3 B - 2 D , (b) 2 D T + B ; 7 . (a) det( D ), (b) det( A ); 8 . (a) C T D , (b) BA T ; 9 . (a) - 2 A T + 5 C , (b) B T + D ; (3) (Inverse of a 2 × 2 Matrix, P. 149 #14b) Let A = a 11 a 12 a 21 a 22 , If a 11 a 22 - a 12 a 21 6 = 0, show that A - 1 = 1 a 11 a 22 - a 12 a 21 a 22 - a 12 - a 21 a 11 . (4) (Singular Matrices, P. 149 #16) Let α be a real number and let
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Unformatted text preview: A = ± α 4 1 α ² , B = 2 α-3-1 5 1 3 α . For what values(s) of α is A singular? What about for B ? (5) (Determinant of an Upper-Triangular Matrix, P. 157 #6) Let U be an n × n upper triangular matrix. Show that det U = u 11 u 22 u 33 ··· u nn . 1 2 (6) (Elementary Matrices) Let B be a 4 × 4 matrix to which we apply the following operations, in order. · Interchange columns 2 and 3, · Divide column 1 by 6, · Multiply row 4 by 5, · Add row 1 to row 3, · Replace row 1 by row 4, · Subtract column 4 from each of the other columns, · Delete row 2 (so the number of rows is now 3). (a) Write the result as a product of eight matrices, including B . (b) Write the result as a product of three matrices, including B ....
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