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2004-2005 Spring MT1 with key - M E T U Department of...

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Unformatted text preview: M E T U Department of Mathematics Basic Linear Algebra LMidterm ' COde : Math 260 : Acad. Year: 2004-2005 Last Name Semester 2 Spring Instructor 1 D t i 29.03.2005 Tiam: : 17:40 3 Questions on 4 Pages Duration 1 60 minutes Total 30 Points 1 1 1 $1 1 1 1 and x = ‘2 U 1 1 333 l —1 0 0 - .- $64: " Name 3 _- Student No I Department: Section Signature : (5+4+3 pts.) 1. Let A = l: :1) Find the conditions on a, b, c, (I such that AX = f: has a‘soiution, i.e-.I the system :1 is consistent. 4 4 4 4 . Q __ E 1 1 . 4044M) fi’fi 01,03xgfl rpm—>0, 01 ’l 1‘0 7 0 1 1 ‘c A/""-> 4 ~10 0: d “VHFWQ. o *1*1-1ld~a ’1 1 1" :q 4 1 4 1'0: 0 *1 00 ‘b'fl Jug—pal 0 —-1 o o lbaa 0414.0 ”“90144’0 O ~1001d-qd-c r 0 0-0 D Id~ +C1- —lo W systm {is Longfbfmlc ”he d +C—b =0 (:7 1 b)Find all solutions of AX n {1) . Horn {"4“ a; “1.9, know ’le1‘ 00'1 411/1‘1 f‘rrqo ‘ 911$"ng __ \0 “WV” 0—100 0 3 000 01.0 QOQOIO __—. 4 '1 “I C)? 1| Q Q 0 @(fifgfiwg #11 I 'l (O K '1 00% (1+(‘3—7F3 . o "a? H‘ 0 ll éfzflgfifif I. OQOIII x12‘ 40010 204’] ’o 30/ XleO 000010 nyfio =2) [gr-XL. Elna-s: 91 51 :1 0) Let Y = y; 22 and B = 1 1 , find all the conditions on a, b, 6,1'311311 that :93 23 D 94 24 1 AY = B has no solution, Le. the sytem i inconsistent. U) Rom W 0i (Md b',-‘“35€e-M+W M. svflem Mn?) is Encorersmt when cebd. 10 —11 11' 0—110 a) Use elementary row-operation techniques to Show that A is invertible, and 11d til—1. (4+4+3 pts.) 2. Let A = 1—“th Draws-4 G "1 ”NO”! 00 “.3 014 “1 110449040 0.10 0—4’10000’1 »—1 “1010064. J fi121; 44.nov, .1 . 1\\\\J .IIIIKJ 111IWA comm 0004 I 11 CL 00,0 0040 0.100 _ 4. 04.0 11.100 001.. , a .1000 _ L U004... T\|\|L 400W— 1.\|.|IIJ q|lJ U040 3 2 0.14.0 .1... 010.0 I «he a 4.41M. 1O fiooow ,. E .110. 4.1 k [I‘l‘L 7“. Tu} will; _, ..L_, s rw. ””0404 0004 0004 ..|.MH.IJ X @rflflwvfi MOOAso 04.40 4.04.0 MMNMDOW J! mOAtOO 0400 01.00 __ FIIIIIL Illltd. mWOOfi 43000 E q A 2:124 .w : VI]. : A vm /. /I_I w ES 2 n... m 4.440 m Que E m 414.1 m all m , , . .I m will}J WIIIJWJ 9. mmmm SUMO m 0001 0601 0001 00.“ mmnm0100 I .1: m. 004.0 004:0 004.0 00»... 001.0 000,, m 0490 0410 0101 0400 h rail. @940 m 1646 1000 4000 00 m X 1,400 m a, z a Jan 1.000 Cum]. .6 E (3+4 pts.) 3. Use the properties of the determinant function to cbmpute the following determinants over R. 89 88 88 37 ..,[‘I1f1_)m 1 1 21 F‘— a) Compute det(A), ifA= [2: Z: :2 S: . n 1 1 11 2 if”? % % m w ‘73*fifflj 4 1 1 I O 0 1 O ' “*7 o O o ’1 8685 34 J; ’1‘ A P MUG U56. [Ckv' Opel-awn!) m‘ ML _ '1LjP-C cfl‘ «(5 ~53 Serz... M, 1% 0p. er m: (ficve NM? 606% efiécr m; M «em-ammr M; M? (3112+ A: (16113 .. 0 0 1 olefib: $101 1‘ ’f H 1 : Sea-36:4 @685 83 3635“ 2 d y 4 2d 2y b) Giventhat det(l:a —1 z:l)=4,con1putedet([—a+Qy 1+dy —z+y2]]. b :c 1 b—2 m—d 1—3; A . . E: , _ A Hr? —~a 1mg ) 41113 Hm Wm?- M“? b 11 '3““1"9F3' km. X-jj 1—3 bie’rn («CF-““7 “def“ “jg? *dei‘fl ”71W”? Q(*d€¥fi)=d€‘fg u ’ “if" “309‘; (38%- rfimum W, “*2; Lu “W: 612‘ ;m, may (M ‘ '. - - c. r: (1?. (What) by OPEQ» c:.r;~»rg~~—>{3 3 H; ...
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