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Unformatted text preview: SOLUT'QMS Make sure that this test has 10 pages including this cover The University of Ontario Institute of Technology
October 20, 2006 Mathematics 1850U and 2050U
Linear Algebra: Midterm Test I Time: 75 mins Name : Student Number : Signature : Special Instructions :
0 Show all your work.
0 Non—programmable non—graphing calculators are permitted. 0 No notes or textbooks allowed.
o If you need more space than is provided for a question, use the back of the previous page. 0 Read each question carefully. Grade Max Rules governing examinations OO 1. Each candidate should be prepared to produce his or her
identiﬁcation card upon request. 2. Caution : Candidates guilty of any of the following or
similar practices shall be liable to disciplinary action: (a) Making use of any books, papers or memoranda,
other than those authorized by the examiners. (b) Speaking or communicating with other candidates. DOONQCﬁrPCOIOHQ
mmcncncncnooyp (c) Purposely exposing written papers to the view of
other candidates. Total CF!
0 name ID. No. l. (8 marks) Solve the following system of linear equations using either Gaussian elimination or Gauss—Jordan elimination. $1 — $2 — 2$3
—2£E1 + 2:102 + $3   

H —:r1 + $2 + IE3 0
l 4( /;l,( lwwﬂ+wwz l “l ’1 1"
“l g 1 (,l f“? o o ~3t’3 .w l I I le+ww1 :0 OD” uﬁﬂ l/Jaokéwlosl’i l/vni'l‘omi 27Mtﬁ‘om Meow;
\ét’ml’lac/l 7/? ¥(’>(2,: ) name ID. No. 2. (4 marks) Let [ —3 0 2 —1 3 4 —1 —2 —1 0
2 1 3 1 2
2 0 1 —1 —2
0 1 2 4 —2 I—_l If C 2 AB, ﬁnd 032 (Le. the entry in the third row and second column of C). 1/2,“? 13L crow/1 1% row 0’? 4 M
2M Colt/(WM 0% 1: 0
7b CEL‘CI O 1’! DJ ~l I name ID. No. 3. (8 marks) Let [321
—1 O 21 3
0
2 0
Find the determinant of A. US? Colmwu/l oqﬂmh‘oms +0 r€ohace ‘11) \COVM JA/‘OAJS 000 A:  H 2 1 wej ’écv CO £¥F4M$COM 9. l D ’1 Q [ 402
AMAM J; l g 3 401.g+¢ol.z J , E 3
9‘ O 1 O (44M? 07;), [(025144 M CD #9
a~l :10 w“szde O‘IFPO MW M81 éoéaofov .eyﬂgw/ﬁfbm paw/7 3rd row: A , Oalo martin—1)“; 4/: (am/(ram name 4. (5 marks) If 2 1 1+ (3A)—1 2 2  
l—I ﬁnd A. T+C3AY¥P 1]
g9.
71> ’3‘? 1—“ ($33”, 1
/I 7 h; I \
77C 3 [a I]
71> —, t \ ﬂ,
39‘ [aw] gum" ;\ \
a; J (— ID. No. f‘ovo 1+ rot/~51
> name ID. No. 5. (5 marks) If A and B are both 3 X 3 matri ssss uch that det(A) = 2 and det(B) = —1,
calculate det(A2BA_1) Recall that: det(AB) = det(A) det(B). whim“): Mmmmzwwm) = meﬁmng) m “9 snug M (N) = J yam—w) = (2)C23 (/l) name ID. No. 6. (5 marks total) Find conditions, if they exist, on the numbers k1, kg and k3 such that the given system has (a) no solution, (b) a unique solution, or (C) inﬁnitely many solutions. $1 —2$2 + $3 =k1
—IE1 + m2 —2:r3 =k2
$2 + m3 =k3
abz/Lxgki [’1lyﬁh,
” I “Q { 92L a? o «I “I $925122
0 I l i 925 D I l b3
i "Q l E 202,!
WM? 0 —«l v] E92,+927_
o o o I: 9zi+92w9§ a) 1,5 ﬂz.+922+ﬂzgqéo (xx/um wig gjglvzw wTLl
Rig/Q “A Comgfs’l—fw'l’ M Mr? will b/Q W0 (,0 [Ugh—(9M
/ l0) WK ﬂ‘f‘z VLO Valmﬂg (375 02\)&_2)92$ SMOM
Mm K egalej OM gblmh‘bl/k Naughty “PK/off Pr f§ vgl‘mvewliloLQ_ 9ch
ﬁr 76 97mm XVUS Ween/:5 M—IL name ID. No. 7. (5 marks) Find the coefﬁcients b, c, and (1 so that the cubic curve y = m3 + bzr2 + cm + (1
passes through the points (m,y) = (—1,1), (m,y) = (0,0) and (m,y) = (1,1). Note that the
coefﬁcient of the m3 term in the cubic equation is 1. Hint: all of the given points have to satisfy the cubic equation. 01 Qamgg [4m @iivvxiL/‘QLEKO Vt _ __ l/\
\f‘ >4 P (I OH?
g
__ y (ﬁt/0";
a“ W .P name ID. No. 8. (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 row operations on an augmented matrix never change the solu— tion set of the associated system of linear equations. i (b) (1 / 2 mark) Elementary row operations on a matrix never change the value of its deter— /
minant. _L (c) (1 / 2 mark) If a system of equations has nontrivial solutions, then the system cannot be
/ homogeneous. L (d) (1/2 mark) If A is a square matrix that is not invertible, then the system of equations Ax = 0 has inﬁnitely many solutions. _/l/_ (e) (1/2 mark) If A is an n X 71 matrix, and AX = 0 has only the trivial solution, the ATX = 0 has only the trivial solution. _'l.’_ (f) (1/2 mark) If A, B and C are n X n matrices such that det(A) 75 0 and AB 2 AC, then B=C._’l/_ (g) (1 / 2 mark) If A is invertible, then A‘1 can be expressed as a product of elementary matrices. L (h) (1/2 mark) If three lines in the :r — y plane correspond to sides of a triangle, then the system of equations formed from their equations has three solutions, one corresponding
/. to each vertex of the triangle. 4/— l: (i) (1/2 mark) If B is a matrix satisfying AB = I, then B 2 A71. (j) (1/2 mark) If A and B are n X n matrices, then (AB)2 = A2B2. L (Wm erg cam \94 My, I? A owl/W E E6 vw’l’ $7MV€ ‘2 éj[IOO (o (o
OO( [0)] l0 :
OI name ID. No. 9. (5 marks total + 1 bonus mark) (a) (3 marks) Show that if M is an invertible matrix and k: is a nonzero constant, then the inverse of the scalar product kM is given by
_1 1 _1
(kM) = —M
k
(b) (3 marks) Suppose that A, C and D are all invertible matrices, and that
A230 2 DA, Solve for B. a) 1? wt show JEMA)()2M) >1: WW Q37)": fez—Mil «All/ultra WL Vl/LOW M/(M: HIM/13f 7> 62 M") (9m) :(t W) (ﬂat/l) ...
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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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