111-2007&2008-3-M10-July2008_2

111-2007&2008-3-M10-July2008_2 - Kuwait University...

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Unformatted text preview: Kuwait University Math. 111 July lst, 2003 Dept. of Math. 8; Comp. Sc. First Midterm Exam Duration 90 Minutes WW Calculators and mobile phones are not allowed Answer the following questions 1 1. (2 Points) Let A: [ 0 3 ] . Find the value 0ft for which A2 — I; : O2, ‘2. (2 Points) Evaluate the follnwing determinant i—‘wm MOP-d l ._i 3. (2P0ints) Let A be an n x 11 matrix. Show that if (Ifl + A)‘1 2 In — A, then A2 = On. 4. {3 Points) Let A and B be two it x n matrices. Show that if A is symmetric and B is skew i symmetric, then AB + ‘28 + BA is skew symmetric. 1 —1 3 0 1 ~2 5. (3 Points) Let A“: 5 1 0 and B: 1 0 1 2 0 1 2 3 0 Find the matrix C such that ATCB" = {3. 6. (3 Points) Let A and B be 3 x 3 matrices such that |A[ = 3 and |Bl : —4. Find I—QATB‘]|. I G 7. (5 Points) Let A : 1 0 l. l. ow.— (a) Find the matrix .47]. 2 (b) Solve the linear system AX : B, where B : 1 [l 8. {5 Points) Find all values of k for which the resulting system has: (i) no solution, (ii) a unique solution1 (iii) infinitely many sclutions I + By + 22 = 4 2:: + 73; + 72 = {l ' 3:: + 10y + kzz = k. 00 99 ' Kuwait University Math 111 July lst, 2003 Dept. of Math. 8.5 Comp. Scl First Midterm Exam Solution 1. AQ—I2:02=>A2:12: gttjg] = 3]:2t+2=0&t2=1=>£:—1. —1 =(O—l+24)—(074+6)=21. P‘WN MOP 3. (In+A)_1=In“A:(In+A}(In—A)=In=:>1fl~A+A—A2=In=¢A9:O n. 4. A is symmetric and B is skew symmetric => AT = A and BT : —B. {AB + 23 + BAJT = STAT +‘2BT + ATBT : —BA - 2B — AB = — (AB + 23 +_BA). 9 T 3 1 —1 3 , 2 6 —6 1 5 2 01k? 5..ATCB“]=13=:>C=(A“1)TB= _1 1 0 1 0 1 : 3 01 2 3 0 e. l—ZATB"| : (—2)3 14113;“ = (—8) (3) ("1) Z 6- 7. 110 10 0 11 0 1 0 0 [#1421210 010~010 2710 011:001 0113001 1 G 0 —1 10 ’V 010E 2—10 ~[135A]- 0 01 —2 11 —1 10 —1 10 2 71 aA-L 2 —1 0 bX:A"lB= 2 —1 0 1 = 3 ~2 11 *2 11 0 —3 1323313233 low—7321 8'277§0~0135—6~0135—8 31019231: 0122—6311—9 00k2—93k—3 Thesystem has: a. nosolutionisz—B b. infiniteiymanysolutionsifk=3 c. a unique solution if k E R — {i3}. 99 ...
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This note was uploaded on 02/23/2010 for the course LINEAR 0410111 taught by Professor Linear during the Spring '10 term at Kuwait University.

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111-2007&2008-3-M10-July2008_2 - Kuwait University...

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