MATH 204 Exam 1, Fall 2011

American University of Beirut Math 204 Quiz I (Fall 2011) Name: ID#: Instructor: N. Fuleihan Re ** Circle your problem solving section number below: Section13 8:00 Tu rion 14 Section 15 11:00 Tu} / Ne Grade y, 2 /18 3 /18 4 /24 5 /20 6 122 Total /100 Time 45 minutes. (6 pts) (6 pts) (6 pts) 2-=1 0 1) Given the matrix A=] 1 -1 -3}]. 2 3 -2 a) Calculate the determinant of A using the cofactors. 2(4)" "Mu -! (-'"* Mio Ja\ = @cu -1e. = ealt ale]. GleaGee Ge . = 2244226 2x, -—x, =0 b) Consider the system }x,—x,—3x,=2 Use Cramer's Rule to find only x,. 2x, +3x, —2x, =4 ets 2! (“eve By be if a> 2 2.1 SS ¢ ~c ——" = oa 5 «to 26 ab y- 73 43 +2 = 2 en a ae 26 1S 20-12 0 00 0 4 0 c) Let B=|2 3 5 3. 4 |Use the previous question a) to calculate the determinant 1 0 -1 8 -3 20 3 9 -2 of the matrix B 2oe-4 2 0 26-10 a iiaingaitic NM Mics 2t¥ 23-5 : 4 2b 5 34 Jeucy- 4D Mayet oe lo -1 -3 lo -( 8-3 - io 3 4 ** 26 Ss 2te 2-10 = 4 (3)cea2 4 (3)-)) Mera 12 ee 9 e 3 +4 = 1% dek(A) = jalab) 6 -2 -3 d) Deduce from the previous question a) the value of |2 -1 —3]). Justify your answer. (6 pts) 4 3 -2 6 -2 -3 3 -2 -3 2 -l oO SA Sloe le «ft 2 22 /ha'S yp =f 2 3 +2 23 -2 Ri- Rp = 2 |Al2 2(2e) (4 pts) e) Find |54| [eal = SP dale 9° (20) 3) The sizes of the matrices A, B, C and D are 2x5, 5x4 , 4x6 and 5x4 respectively and O is the zero matrix. Give the sizes of O and of each product if it is defined. vovede BX Bree 2A BNAK 0.9. HL BBS, malix AOB & axy abr aaceseocestaavenesed vere! mat..aleprnaol Su Y¥xg Sx¥ BCO.......... sz ‘ec or Oa COL “traliix {360 le xn Sku XG 6¥T (10 pts) i+3j if i<j 4) Let B=(6,)bea2x5 forwhich b,=45i if i=/ 2-j+? if i>7 Determine 5,, and the entries of the first row of B’. big = 14.3l4¥)= Ke a by ions 13 by, be (6 pts) ij “ bia bis big bis ues 2 bar Bs - o = bia bz bay ber be3 bay bas biy ae bis bes bu = slic bar= 2- «ge2 4 44UaS ; — [big 213) bis$ bez 8 5) Solve the matrix equation x+1 2x+1 x+5 3] y-2 |-2] y+3 |=| 3y42 (8 pts) z+4 re 4 xt Yx+2 byte aye] \4y46 by 33 4! 2 4 Bxt3 -4¢x-22%tS —? 3y-6 -W-6= 34+t2 >» 4 —1223 442 aye —>ly=4 saat - eh =e ae 1424 —9/3=-'2 A wee lets 9-2 = 4 —x2-2 } 6) For which value(s) of m can you apply Cramer’s rule to determine the unique solution x, +x, —-x, =1 of the following system } 2x, +3x,+mx, =-2. (8 pts) x, +mx, +3x, =2 By Camus Rule the Aislin Bas A unecgue Arlution if p= [23 >| #c iol [a -l “— 2 % Ns 25 =i fs af | -: Im>d = Gent) — (6m) ~(om) = (F-w)-6@ +m- Mme = (4-*) -3-™ Dto — (Ss) (rm) 40 7 ma -3 adm $2 7) Determine the number of possible seven-digit telephone numbers if none of the first three digits can equal zero and: a. The inst digits $ CU 8) cL ley ey os eh 3 loxtoxiox| = Gxld — 9x4x4 xloxioxiox|l = 4 Sy v= b. All digits must be even ty uh ity =e Sy es ?, 6 3 c. No digit can be repeated oy ums Yoo oa 9x¥ x HXFRKEXTXY (22 pts) -1 -1 2 8) Consider the matrix A=| 0 -1 -—2]Find A’! using the Gaussian elimination. 2 0 -3 Hj —*Xy 42%, =3 Then use it to solve the system: 4x, +2x, =2 2x, —3x; =2 4 1 2 onl S24, 96! WO!) oiRe i j) 2 Serko o-29.' 201 o-2 bt . 201) R422 Rk. 110 1 o-4Yicrt § 0 Riad ks | o-4)~ o 1k fo tO | Ry-2Rs eye ye o | +2 -% + oo st rt / sks - 75 zs WY 3/5 “3/5 “fs "a Ws \ 3ls 3/5 4 -2 _ ; “Layee US Awa Ale Vs UE ol a 4 - <4 YS" "As -Us Vs o © :