sample test 1 solution

sample test 1 solution - MATHl850/2050 809 Name: SOLUT LOMS...

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Unformatted text preview: MATHl850/2050 809 Name: SOLUT LOMS Page 1 of 6 (1) (6 marks) Solve the linear systems associated with the following matrices: \ X1. X5 XCI XI K7. K3 ® 3 0 —1 4 CD 3 —2 5 (2') 00®23 (ii) 0®2—2 0 0 0 0 0 0 0 (13—1 (L). L€T\xz=$l\><u=k5,wwm she‘ll Q2) R3: DE“ “‘0” Q9“ " x5: 73—17“ Va: X2: ~Z ’sz FILM K -. 7n: H ~3xz+xq (Ll: m: S¢5><1+Zx5 So lXfi— LV$5+£5 So XV‘?) (2) (5 marks) Find conditions on b1,b2,b3 such that the following system is consistent: 331 -- 2:132 — 3:3 _ b1 2331 —— 5332 — 3333 = b2 3331 —— 7332 — 4333 = b3 I 1 ~\ lo} 2 ‘5 ~35 b 3 7 —q ‘03 \ Z —\ lo R: ~22 ' 13:4‘ 0 l -x “.2101 23:23-3L O \ ~\ bs—slm \ 2 ~\ 5, ~—? [Z513- 2.1, O O 0 ‘03“l01"l0\ NEED log—lofLJfiO 47oz. sysrem 1'9, (5;: CQuSLST-E-N—r MATH1850/2050 809 Name: Page 2 of 6 (3) (5 marks) Find the inverse of the following matrix (make sure you label all your row operations and state What 14—1 is explicitly): 1 0 1 0 —2 0 —1 0 A 0 2 0 0 2 0 2 1 [A 1 _ \ o l 0 l o o o “ ~L O -\ O o \ O o O 2. o O o O \ o 7. O 2 1 o o o \ l O 1 O l o O O Kz‘Kfi-Zfli O, O \ O 2 l O O ‘4 \ LHLQH-Z£\ O Q 0 1 “Z o O l l O \ O l o O 0 {11,995 0 x o o o o y» o ‘9 o o 1 o 7, \ o o O o o I —z o o l l O o o ——l ~l Q 0 RV. R i—Q—z O L Q o Q o 12- O P [fix—1‘3 5 Q o \ o 2, x o 0 V o o O l ’2, o O I MATHl850/2050 809 Name: Page 3 of 6 (4) (5 marks) For which values of k does A fail to be invertible? 1 —2 —5 A: —1 k 6 2 —4 k+2 A MOT Quart-(gee {PF MA-éo. \ — 7. ~ 5 \ —2 —S M A s “\ k G h 0 \<r'L l Lgr K1 1 —“t (on, o o \¢+\z K3’ZLi : QU-Lka) :0 \¢—.2 0L ks—lz. S0. A is swéumo. \IFF k=z OR. Ltfllz‘ (5) (4 marks) Solve for the matrix B in the following matrix identity ( show all steps, and mind the order of Operations): 2 —1 TAKE \NV€%€5 goTL—I— slbgs: A— T _’ \ _7_ .2. g ’——\—2. 7' Z ’5 ADD :2 To ,go-ru $1665: _\_ gr: 1 -2] D” 7. ~Li SCALAIL MULTl-OLY ray 1 ’goTH- srozsf [$TT[UL «Li-X q '3 TAKE TUMsooszs (50TH 5‘13“: (5 [M HA) MATH1850/2050 809 Name: Page 4 of 6 (6) (3 marks) Find the second row of the matrix product AB Where 0 —5 0 2 3 3 1 —4 A = B = 2 3 0 5 3 0 2 0 1 —2 —3 2 Apbs[zo ’6 toli5=[0i ——\ 1 \ol] (7) (3 marks) Find C42 (the 42-Cofactor) for the following matrix MATHl850/2050 809 Name: Page 5 of 6 (8) (5 marks) True / False. Indicate whether the following statements are always True or sometimes False. (a) Every homogeneous linear system is consistent E/True False (b) If A, B are square matrices of same size, then (AB)2 = A2B2 True E/False (c) Every singular matrix has at least one row of zeros in its reduced row echelon form E/True False (d) If A is a square matrix with two proportional rows, then A is singular E/True False (e) If the linear system Ax = 0 has infinitely many solutions, then the matrix A is invertible True 3/ False (f) A linear system of 4 equations in 3 unknowns must be inconsistent True 3/ False (g) The permutation (4, 2, 5,3, 1) is odd 3+L+z+\ =— 7 Hive-Lake“ 3/ True False (h) If A is an upper triangular matrix then AT is lower triangular 21/ True False (i) If d€t(AB) = 27 then A and B are invertible V \F A“; saw 3/ True False (j) The product of any two elementary matrices is an elementary matrix True 8/ False MATHl850/2050 809 Name: Page 6 of 6 (9) (4 marks) (a) Show that if M is an invertible matrix, then _ 1 _ det(M) [Hintz use THM: det(AB) = det(A)det(B)] det(M_1) (b) Suppose that A and B are invertible matrices, and A, B, C', D are of same size. Show that if A‘lC'B = D, then C’ = ADE—1. A“c%=b «a? Aé‘kgyg‘: mm" =’ CM")C(%V>")= Ame” =7 10:: AM" => C: Ame" BONUS (4 marks) Show that if A is an invertible n X 71 matrix, then det(adj(A)) = [det(A)]”_1. A“ z fim cam-A ca 0%: A": 35%;; 945A) :D ~_l_ 3 )V MA (MA 3) OQJ : wAywyfl—l : (fie—Q70“ ...
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This note was uploaded on 11/10/2009 for the course MATH 1850 taught by Professor Mihaibeligan during the Spring '09 term at UOIT.

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sample test 1 solution - MATHl850/2050 809 Name: SOLUT LOMS...

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