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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
23:233L O \ ~\ bs—slm
\ 2 ~\ 5,
~—?
[Z513 2.1, O O 0 ‘03“l01"l0\ NEED log—lofLJﬁO 47oz. sysrem 1'9, (5;: CQuSLSTEN—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‘KﬁZﬂi O, O \ O 2 l O O
‘4 \
LHLQHZ£\ 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 MAéo. \ — 7. ~ 5 \ —2 —S
M A s “\ k G h 0 \<r'L l Lgr K1
1 —“t (on, o o \¢+\z K3’ZLi : QULka) :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 ,goru $1665: _\_ gr: 1 2]
D” 7. ~Li
SCALAIL MULTlOLY ray 1 ’goTH srozsf [$TT[UL «LiX
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 42Cofactor) 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 HiveLake“ 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 camA 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.
 Spring '09
 MihaiBeligan
 Linear Algebra, Algebra

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