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111-2005&amp;2006-2-M10-April2006_2

# 111-2005&amp;2006-2-M10-April2006_2 - 5/5" Kuwait...

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Unformatted text preview: / \ 5/5?" Kuwait University, Faculty of Science W Dept. of Mathematics & Computer Science LINEAR ALGEBRA (MATH 111) First Mid-Term Test April 3, 2006 Duration 90 minutes Calculators, Mobile phones, Pagers and all other mobile communication equipments are NOT allowad. gypsy ‘3’ 59”“ ”MM (‘ w c Answer the following questions: \$071"? 5!? {RM t- H bat" 1 2 —1 12 9» \$953793 ”EM" 1. [3. pts.] Let A : 2 —2 and B : .lf possr le compute . _1 0 D —1 3 /a.) (A + 231")?" ,/(b) AB —2 8 ~63 2. [3 pts.] Let A = 3 2 5 1 O 7 ((a) Find the matrix B in the reduced row echelon form that is row equivalent to A. I (b) Is A singular? Justify your answer. X3. [4 pts.] Find all values of k for which the following linear system m+3y+kz 4 233+ky+12z 6 digit.) has inﬁnitely many solutions % has no solution (c) has a unique solution. 96: Solve the linear system for k = 7. wiggfzg \$5! Mka W 4. [2 pts.] Prove that the inverse of a nonsingular matrix is unique. .ccc \$5.[5pts.[LetA= 2 2 0 l O l 4 1 t(a) By reducing to a triangular form, ﬁnd det(A). 7‘(b) Find det(2A‘1). / 6. [6 pts.] Answer each of the following as True or False (justify your answer): i (a) If an n X 71. matrix A is symmetric and skew—symmetric, then A : In. [Ly/a" {(b) If A2— 0”, then the inverse of I1.L + A is In - A. X mail If Imam x (d) The permutation 45123 on the set {1, 2 3 4 5} IS even. /7. [2 pts] Show that if X1 and X; are two different solutions of AX: B (B 75 0) then 1X1 + éX2 is a third solution for this system. OO 99 1 -1 +2 “2. ((5 __é "I O O , ‘2‘ OK“ (A :- 15) ‘3. '5 mm 0 ‘ O :8 x o "1 -O O ‘ ‘0‘; Q {S (”cw-J Cia»;\\)0n\€»)~' E012} ’ Q \s Msiﬂuﬂgc. \ a, «c '1 L; if“ ‘ ‘5 K Lr ] [ '_ c9 K-6 —-m¢-e") : .-1 an ‘9 LC? 6 ’ Veg SSSV-Gm NS ingmd’ekk New“ ﬁﬁJao’V‘xs. ‘m \9 DC: 6; ) V86 833%va 94M {\0 Sokdncm \ "=2. =1 ‘. k+ ““5“— ® 0 V3 ‘ ‘0 [ \w [ogun-y The. Saw ‘6 W: Eng—4'34: -9.+1_b k; “3'1“ 5%;va (Te/2: F. '1' '- t‘a—aﬁ (ngéf _ “‘43 § 0 ,\ 1 ‘ O "-l '2 1 O ._( 1 5mg C —t L -1. _ O -\ '9.- -7_ ‘3 C) -\ 9. ,1 '2. a o \ ' 0 ’2. L45 0 o 6 1, 0 ‘ ‘4 ‘ 0 \ L+ i c; O 6 .—l l 0 ~\ ’1 o .-\ m «7. ‘ 7" = —36. o ‘5’ 6 ,1 o O O 6 m \u‘x‘W— 6"pr 3:. = .. LE2. 9‘ ‘56 c — -— 9'30“ 5; 0‘1 WASL ‘ (\‘mchPx 112) (9.1.9.2) SEQ—On .30 . Ln-_ 9. ”333;; Cia+<>dv\$n §\\: -3_“};_g;&_.vg_, : L“ 1 (in- CH Vindm :3 *m 1‘», {\2 1“ Q“ Twat 693:0“ .——_-.7 \CAR‘\ \-: \Q)n\ —._—_j \{A\=o '57 \wa CM “Woe '°'. wiﬁ; gala; 6 m vars-(mg :3 ewe/n ‘ -:-—__./__/ 1 Q 05% + imam av. m Muxa— “KCMM * 29m» 55f gﬂ 7P ﬁlmigcgw JAM/my , 5 m W M4 f M Ma A+,4T {g gamma/fl‘c ”is ”‘“‘/A+/ﬂf= Aggy: @(53 A , A7 I“: g szu sammf/fa‘ éjwh ”b” 12%! ~7~-~HV_..,, Mm,“ fa) MM") « #1 (NSF [42,4 5))4,1~\$ym ‘ :ﬂT+/4 ﬁr]; ”41 3314!} iii” : Ar 4 705’” “/9 A Mfr H, WWW/W! C 7v: ””””” r, _. {A ~47”) I, + “F Mir/”Li , {/QHEV’; A «B @M’éﬁj‘g «ﬂ “RA-f ' m) , r \AT 6* (”7“) ’ ‘ ”(S how (”ML by 141519 N 4 ﬂmmn wgém Wm/ ﬂ!“ l/WM 721a“ ¢v1~€ {aw/“z WW. {1' ﬂ“ {Wit/21% 19 Many {MM/WW [HM], I ’% 1"] M0! M4. any“? fail/{WAX (m Sf/éw W : (“U-Mu: £51, , \4/11le r+§zl [LG )‘f 5’ (J. :3 AU. :‘0 M1 K (A (J7 3) A [/7 : (9 : Pb'f' S 19 :(YV4’§>JD u/ ”” ﬂ[ru:+suzj : w AMA—«4W :97‘W4'gl/{1 {3 45M. "3? r611 1 (mm 99 ...
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111-2005&amp;2006-2-M10-April2006_2 - 5/5" Kuwait...

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