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# exam_2_old - um‘nsibth 1(15 pts Consider the following...

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Unformatted text preview: um‘nsibth 1. (15 pts) Consider the following matrices: T I A: 3X{ 4 7 4 3 7 3 A:(12>B:(127)C¥(6) 0%3’5 5 G 2 0 4 1 C ’ 3x! _943~6 ‘15? D52)”, “<21 7 5> “(:32 E 53x3 \Vliat are the dimensions of each matrix? T Col snug Perﬁn‘m the following operations: «a d (a)E+B (l))2><D ((‘)CT><A (d)(let(E) I.)- (R) Hy 5’15 5+7 __ g- 3 ,O Eta: 7*! 21-], :*‘7 ’ L a [O b 0 M a, 6 ﬁn {) ZXD- XX?Z)Y “37‘?- 2’6 ]=( V -L HI ID 17;; 2x00) 2"? 2Y7 (c) T ‘l 7 elf/2+6) l 1L : 'Z‘fﬁf‘y CW4: (3 a :)x(f C)( r: (23 3?) 0‘ I J' I 3 .7 2 ()d‘+7 1.3 :(l)l1’+(-:)';f+7q.) ‘1 a 5 a l» W z—ﬁ-é‘l= "3"2 I2 _:(q2-1|) +X(°Y)=' I 2. (20 Ms) USU Cramer’s rule to find Lilo inverse of the matrix. \ , q ‘- 1 Z '- ’ Find the condition number for A using the Euclidean norm. I 9 ° l 2 R I 2 J : := ) 3 ‘1 ‘ 3 ‘1 c o o ‘2. 1,: "‘=-‘ C(‘qwcr‘J (2*1C ":2" J3; - 531-57“? UN i ' ‘l 2 q- (is «- 5 '2_ 3'1 ’2— u u C : T a -2. I {3/5 -‘lb) Ckcox "_ I 2) g o , I ° AA - (.3 ‘I (a l 0 ‘ H4“: VIN-(+9141: = Yso [M4] = m -: VLLeLgfii-l ._.. [33 co«1(4)= mm- MW” = “‘7‘ T- ‘2“ 3. (15 pts) Use Gauss elimination to solve the ibllowing set of linear equations. Show your work! :I'] + T2 +13; : 3 2T2 + .173 : 6 37170132 -— 2.1‘3 :1 4. (25 pts) It is possible to doc<:>inpose the matrix A into the following LU dominposition. Using this decomposition, ﬁnd a solution to AX:B Where LUX==B at 9-49”? B_ 1 [:“z‘Lj u K Solve— ) R“ 3 Sal-JN-x tXI'ZwH . a L? I B What is the rank of the matrix A? What is the uniform vector 1101‘111 of A? Is this (ioconi )osition consistent with the CIT ut moth) 1‘? n“, I ’ “ \> Ms): .41 MPH} 3 L911“ 114-6) (lb oak-v. Hf'f/ - . V: I usL 2' g g) 3‘ 4-1 ) go“. r .L (25 pts) Use the power method for the following eigenvalue problem, AX’AX: J 4 ll) .1] .l‘l 4 10 () 1'2 : /\ 1‘2 0 1 .773 1‘3 Assume an initial guess of X (0) : [ 1 1 HT and perform two iterations to ﬁnd an estimate of the largest eigenvalue, 11.6., find X (1) and X (2). Use your estimate for the largest eigenvalue and eigenveetor to set up a matrix, A2, Whose largest eigenvalue (AZX:/\2X) would correspond to the next largest eigenvalue for the original problem (AX:/\X)‘ but do NOT solve this new eigenvalue problem! (‘0 (9 i .956?— XLL) .. l L{ t0 0315);): (\$3333 I; ‘2 7 0.738 w 7)” w New,— “’3‘”. ‘1’)- 0,5711. A = A - 20.) Xu.) X(;)T X ’ (O.‘3‘7 > L 0.5””! q '0 0‘57“ . zu 05/27 A :( i1 lo 0 - '33} 0,530. (0.77“- 0 5 ) L 0012—! .. o “.4151. 51.3%)? 3.7372. A'; ‘- ‘ :‘o to - ( \$7708 S-‘IB‘ID’ "(.:"")’a to <7 I {@371 ‘05.“ 3'5“) 6‘2 (\LLY ~45509 ~75?» (I‘M-AL), - . E‘I-sdl' ...
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exam_2_old - um‘nsibth 1(15 pts Consider the following...

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