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111-2006&amp;2007-3-M10-July2007_2

# 111-2006&amp;2007-3-M10-July2007_2 - Kuwait niversity...

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Unformatted text preview: Kuwait niversity - Math. 111 .lluly 2nd, 2007 Dept. of Math. 35 Comp. Sc. First Midterm Exam Dur'ation 90 Minutes l Calculators and mobile phones are not allowed . ,1,“ MN Answer the following questions a) 61”: gm “L Air/\— (5 gym. T ‘ M‘u/ 1 2 -2 @ 51”“, [by A 4“ 151%,... . {2 Points) Let A 2 4 1 0 f/I D —2 1 z; Find a symmetric matrix S and a skew symmetric matrix It” such that A n S + K , / 2. (2 Points) Show that if the linear system AX = B has two different solutions, then it has inﬁnitely many solutions. Mm Points) Let A = [ i g ] . If possible, ﬁnd a. matrix B such that AB = I2. ,1- 4. (2 Points) Show that if A and B are 2 x 2 skew symmetric matrices, then KFind A and B if 6. (3 Points) Find all values of a for which the resulting system has: AB : BA. 1 U 0 l l] (i) no solution, (ii) a unique solution, (iii) inﬁnitely many solutions \$+9y=a so + egg 2 3 . 0 1 2 1 0 2 4)»! fr. (3 Points) Let B = _ 1 0 0 and 6—1 = 0 —1 . 3 . Find BC. cac _ 0 _2 0 1 1 0 : g a 4} /§l (6 Points) Answer each of the following as True or False (justify your answer): if 2 3 E 6‘ ‘ -' l ( (ey/I/f A is a. 2 x 2 matrix such that AAT = 0, then A = 0. T A4: 77* A'A'T’" 9““? (b‘D/ll‘he product of tpvo diagonal matrices is a diagonal matrix.“r ; I (c) If A and B are nonsingular matrices, then A + B is nonsingular matrix. F. (dyIf A is a nonsing‘ular symmetric matrix, then A‘1 is symmetric matrix. AT: A n» (/l U“ (M? :(A)’ , Swwhw \$14-: a ’4 ll OO 99 Kuwait University Math. 111 July 2nd, 2007 ept. of Math. & Comp. Sc. First Midterm Exam Solution 1 2 —2 14 0 7.014 J, 1. A: 4 1 0 =>AT= 2 1 —2 ’fL/A1ﬂ‘7)+%ﬁ4-"‘) 0 —2 1 —2 0 1 ‘1' 1 R 491:, 5 1 3 —1 0 —1 —1 S=%(A+AT)=[ 3 1 —1:l,K=%(A—AT)=[1 0 1]. —1 —1 1 1 —1 0 2. If U1 and U2 are two solutions to AX = B such that U1 % U2, then TU} +(1 — 7‘) U2 is solution for every 7' 6 IR. Thus, AX = B has inﬁnitely many solutions. p2 H(Sﬂ/rH’/-1 :sﬂu, Hw‘luz: 513+r K 2' 3. If B exists, then B : A‘l, but 1A is singular, so B does not exist. v-‘aﬁul' Niz H”) waquﬂﬁr (I W 0 a b —ab 0 4. LetA_[ﬂa 0]andB—[_b 0],thenAB—[ 0 _ab]—BA. 1 0 1 0 1 '4’”;lérl9'1ﬂ*”83l':.,: a .H .=1 -1 m ~+ ——~> The linear system has . a 6 [AzB]= ~ 1 a2 i 3 0 0.2—9 3 3—0 (i) no solution if a = —3, (ii) inﬁnitely many solutions if a = 3, and (iii) a unique solution if a ¢ {*3, 3}. r 102E100 10 23100 7-[0‘1313]=0—135010~01—330—10~ 1103001 01—23—101 10 22100 100'23—2—2 01—35 0—10 ~ 0105—3 2 3 =[I3FG] 0015—111 0015—111 _ 3w2—2 012 34—2 —545 C: —3 2 3 :30: 100 -.3 2 3 x 3—2 —2 —11.1 020 —111 —646 OO 99 a2+b2 ac+bd ac+bd C2+d2 £11 0 0 51 0' 0 G2 0 0 b2 0 O 0 an O 0 bn 61b} 0 '0 O (1.252 0 AB = _ 2-} AB is diagonal 0 0 club” 1 1 C(F) A=[0 (1)],B:[g 1]:>A+B=H inn" I817” ipgrs'isp A and B are nonsingular, but A + B is singular. d (T) (A‘1)T = (An—1 = A“ =1» A is symmetric. O. CO ...
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