fall o6 midterm with solns

fall o6 midterm with solns - SOLUT'QMS Make sure that this...

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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. DOONQCfir-PCOIOHQ 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 V-l/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.

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fall o6 midterm with solns - SOLUT'QMS Make sure that this...

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