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Fall 2010 Midterm #1 Solutions Mihai

Fall 2010 Midterm #1 Solutions Mihai - MATH1850 F10 Name...

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Unformatted text preview: MATH1850 F10 Name: Page 1 of 6 (1) (6 marks) Solve the linear systems associated with the following matrices: 1 2 0 —3 3 1 —2 —3 2 (2') 0 0 1 4 —2 (n 0 1 2 1 0 0 0 1 5 0 0 1 —3 LET ix-Lst- i Xz=‘3 TREK) ~XK5g‘ X3= -'L 404“ 50m x\= 'x,_= A—zxs So 795:7 xl= 9.40.ka 3XJ So— so— (2) (6 marks) Find conditions (if any are necessary) on b1, b2, b3 such that the following system is consistent: $1 — 2362 + 2363 2 b1 2:61 -- $2 — 6363 2 b2 —2£E1 -- 9:62 — 14363 2 b3 1 ‘2 1 L3] 2. 4 '(a ‘01. —z a 4“ lo: I —7_ 1. ‘0‘ Raf-21' “4 o S *lO ~Z‘or\"07. P—gfig—‘LL 0 S' "(O Zbfl'bs \ —7_ 1. 'o\ o g "\0 'Z‘a)+‘°L 22‘ 9—; '91, o o 0 Lu,‘ 4°39; \ -'L ‘7. l°1 T) o \ —7_ 17::- b‘+—§lo1 2'7" ‘5 O o O Lem-57,1405 MEEB L€\>\—bz+los =0 (-212. 519104 To 86' ConsxéTEHt’ MATH185O F10 Name: Page 2 of 6 (3) (7 marks) Find the inverse of the following matrix by row reduction (make sure you label all your row operations and state What A—1 is explicitly): 1 0 1 0 —2 0 —1 0 A: 0 2 0 0 5 0 5 1 _ l O \ o l o o o [1:13:2—[13 o o \ o ‘L \ o o 9.3,‘31 o \ o o o 0 ho Lam—s1. o o o 1 -—s o o n Ital} MATH185O F10 Name: Page 3 of 6 (4) (7 marks) Determine the value(s) of k for which the homogeneous system AX = 0 has infinitely many solutions. [State criteria used, and explain all steps for full marks] 2 —1 2k A: —4 16—1 2 6 —3 1 A”: 9 HAS oo- nAw Seem-ions ice MA=O. 7.. -\ AW\ 0 L—s so::z\(l:;l:+'zc, = 1(k—sDCx—Qk3 =O'tQQ la; 92. hi. 0 0 L— 6‘4 2. 21’32 =9 uAsw—muy wcmsfi kflsozkgm (5) (7 marks) Solve for the matrix B in the following matrix identity ( show all steps, and mind the order of operations): (312+gBT)-1 = [2 ‘3] 3 —5 TAKE \uvetcfis = 35:14, J— (5“: (-3: 3 ADD —2>\:.,_ 2 lg = 2-‘131 -S smut. Mumbw 67 2.: (gr = Tl-[i £1 --'-0 " TLA flags : MATH185O F10 Name: Page 4 of 6 (6) (4 marks) Suppose that A, B are square matrices of the same size. Prove that if the product AB is invertible, then so are A and B. A2> is New?“ => AwCABM—O =9 chfiAJfl$7fio =9 M-fioz‘ New :2 Anbmz \Qweewre (7) (5 marks) Use det(A) = 90 and Cramer’s Rule to find the variable 3:4 for 0 1 0 0 1 0 0 —2 5 0 3 0 the system AX = 0 Where A = 1 —2 3 2 0 —1 —1 0 0 1 0 L 3 J L 4 —4 0 —3 3 J (NOTE. no marks will be awarded for any other method) “AR 8.— 94% A ‘1 CE} 00 g 2 C ((5 )(qf- = — _ = VS" 3‘ "l '5 = "' - «MA “ .k 3 2 f? 2 l H w o .5 3 MATH185O F10 Name: Page 5 of 6 (8) (8 marks - 1 mark each) True / False. Indicate Whether the following state- ments are always True or sometimes False. (a) Every elementary matrix is invertible E/True False (b) If A, B are square matrices of same size, then (A7L B)2 = A2 + 2AB7L B2 True 8/ False (c) If A is square and Ax = 0 has only the trivial solution, then det(AT) = 0 True E/False (d) If A is a symmetric upper triangular matrix, then A is lower triangular 3/ True False (e) A homogeneous system of 4 equations in 7 unknowns must have infinitely many solutions E/True False (f) If A is a 3 x 3 matrix with detA = —2, then det(3A*1) = —3 True 3/ False (g) If A is invertible and B is obtained from A by performing exactly one elementary row operation, then B is invertible 3’ True False (h) If A,B are square matrices of order 5, and B is obtained from A by reversing the order of its columns, then det(B) = det(A) E/True False ...
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