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Unformatted text preview: Make sure that this test has 8 pages including this cover The University of Ontario Institute of Technology
October 7, 2008
Tuesday, 8:00am Mathematics 1850U, Section 002
Linear Algebra: Midterm Test I Time: 70 mins Name : Student Number : Signature : Tutorial Instructor: Special Instructions :
0 Show all your work.
0 Non—programmable non—graphing calculators are permitted.
0 No notes or textbooks allowed. 0 If you need more space than is provided for a question, use the back of the previous page. 0 Read each question carefully. Rules governing examinations Q Grade Max 1. Each candidate should be prepared to produce his or her 1 7
identiﬁcation card upon request. 2 6 2. Caution : Candidates guilty of any of the following or
. . . . . . . . 3 6
similar practices shall be liable to d1sc1phnary action: 4 6
(a) Making use of any books7 papers or memoranda7 5 6
other than those authorized by the examiners. 6 5
(b) Speaking or communicating (in anyway) with other 7 4 candidates. Total 40 (c) Purposely exposing written papers to the View of
other candidates. name ID. No. l. (7 marks) Solve the following system of linear equations using either Gaussian elimination or Gauss—Jordan elimination. l \ ’4 3 L "l Z O .4 l "l I Jimpz L 1
1 Wow; ’* o \ $1 + $2 + 3$3
2m — $2 + 3ZE3 “RWI ¥V‘D\AZ.
————/9 \r‘owl Z
l i—V‘ow'; 3
0
1 g l \ ’3 K Z mwlkmwg C) l 1 A Z o O O 0
3 CD 34‘ amok )6; awe l—Qodivj Wm‘awg <5 )6; Pg wM% (‘3 mow mu Mac/(.4 9UJO¢H 'I‘LAHsL/L \.
67w @ Way/wag 161+f7v2. =’> 167/: Adi:
M Q7m@ (“come ’X,+‘(9~~'(5)f’§1c7=’§ name ID. No. 2. (6 marks) Let 2101 13120—1
A= B= “1‘2”
1_1’ 210010’ 10—121—1 If C 2 AB, ﬁnd 023 (i.e. the entry in the second row and third column of C). Use as few computations as possible. name ID. No. 3. (6 marks) Let Find the determinant of A. M92 (boxcmohau/ eypamgc‘ovx aim/j M \rvw Q21 C’M T Q22, Clixk 0‘74 CL; 4— quk (2+ 1(0) Lu LO) Cw ‘° CO) (23140) C14 name ID. No. 4. (6 marks) If name ID. No. 5. (6 marks total) Find conditions, if they exist, on the numbers ()1 and ()2 such that the given system has (a) no solution, (b) a unique solution, or (C) inﬁnitely many solutions $1 —2$2+ $3 =1
—£E1+3:r2— 3ZE3 =2
I2+bi$3=b2
[v2 1 l \ “l i l
,k 3’3 ; ToutLaw: O l ,a '2
O l E; laz D L lo, l71 ‘uA WK Cage MAW»? QM?— 3 (Mf‘vj
katko/QWS $0 “Awe. K a wwljkﬁ. 4a/wlﬁ‘om 0L) (DH—Z: O HAL W 2.7MH'QM
VOc9LA\A M be “Db/g
’U‘Q wowd h av“ \‘chomﬁfﬁ’uﬂl’ 93${7W «AAA “Km/e
wouLA W M ébL‘sA/k‘OLA o>1<ﬁ \ol—J/L: Q g lﬂzvg;€> mm
m Cvﬂ'wm wouLOK 17/2 csnggkax/JL‘ M MML WOMW \pr but; 2, (9—0433
Uau/‘rml/LLef (y, M \C—U\ /S° MVG Wot/yd bﬂ Tukaﬂwlj Muj (bkwh‘oug name ID. No. 6. (5 marks total) Indicate whether the statement is always true or at least sometimes false by putting a T (true) or F (false) in the space provided. (a) (1/2 mark) Elementary column operations on an augmented matrix never change the L (b) (1 / 2 mark) If a system of equations has nontrivial solutions, then the system cannot be homogeneous. —l solution set of the associated system of linear equations. (c) (1 / 2 mark) A linear system of 5 equations in 3 unknowns must be inconsistent. F— (d) (1/2 mark) If A is invertible, then the system of linear equations AX = b always has F one and only one solution. ’— C— (e) (1/2 mark) If A, B and C are n X n matrices and AB 2 AC, then B = C. L (f) (1/2 mark) Let is H
MOH
HQOH
GOO det(A) = 0 L (g) (1/2 mark) If A and B are n X n invertible matrices, then (AB)2 = A2B2. L (h) (1/2 mark) If A is a matrix of any size, then the matrix product AAT is always deﬁned. _T_ (i) (1/2 mark) Every matrix can be transformed into reduced row echelon form by per— I forming a sequence of elementary row operations. (j) (1 / 2 mark) If the reduced row echelon form of the augmented matrix [A b] has a row of zeros, then the system of equations AX = b is consistent. i name ID. No. 7. (4 marks) Show that if M is an invertible matrix, then the inverse of M 3 is given by (M3)_1 = (M1)?)
3hrer A: M3 B=QVW>3 MW HQ BA=I 4445M Exgm mwmch/l : WAM/k Mal) ‘03 Ami—M 65L Pot/06v”; a’F’ WMQC ' WM" (WM) M M by “$S°¢"“v°‘"“ 1”" “C WHHC leh.P LTcaHow b [‘14 M4 J: M M Manse M i; FHveer/Qe “ I! (Mm—«33
Z M ’M M M “‘3 mﬂﬁrieiﬁi‘j
.3 MV‘QVI“M)M \97 Q§§OCF4+fuTL7 L Mq r M (47 M‘var ...
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 Spring '09
 MihaiBeligan
 Linear Algebra, Algebra

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