Final Examination. 21 November 2012 Tyd/Time: 120 min Volpunte/Full Marks: 50 VAN/SURNAME: VOORNAME/FIRST NAMES: STUDENTENOMMER/STUDENT NUMBER:

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1 UNIVERSITEIT VAN PRETORIA/UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS WTW 16 Lineêre Algebra/Linear Algebra Final Examination 1 November 01 Tyd/Time: 10 min Volpunte/Full Marks: 50 VAN/SURNAME: VOORNAME/FIRST NAMES: STUDENTENOMMER/STUDENT NUMBER: HANDTEKENING/SIGNATURE: TELEFOONNOMMER/PHONE NUMBER: PUNTE/MARKS A 1-15 /15 B 16 /5 B 17 /1 B 18 /10 B 19 /6 B 0 /3

2 INSTRUKSIES INSTRUCTIONS 1. Die vraestel bestaan uit twee afdelings: AFDELING A (Vrae 1 tot 15) is veelvuldige keuse. AFDELING B (Vrae 16 tot 0) vereis geskrewe antwoorde.. Rofwerk vir Afdeling A kan op die blanko papier agter gedoen word. Omkring die korrekte antwoord vir die veelvuldige keuse vrae op die vraestel. Vul dan jou antwoorde in op die rekenaarvorm met n sagte potlood. Beide die vraestel en die rekenaarvorm moet ingehandig word! 3. Vir Afdeling B moet all bewerkinge getoon word! Jou antwoorde moet in die spasie voorsien op die vraestel ingevul word.. Geen werk in potlood of rooi ink sal gemerk word nie. 5. As jy korrigeerink ( Tipp-Ex ) gebruik, verbeur jy die reg om die merkwerk te bevraagteken of om werk wat nie nagesien is nie aan te dui. 1. This paper comprises two sections: SECTION A (Questions 1 to 15) is multiple choice. SECTION B (Questions 16 to 0) requires full written answers.. Rough working for Section A can be done on the blank page that appears at the end of the question paper. For multiple choice questions, circle your choice for each question on the question paper. Then fill in your choices on the computer sheet, side with a soft pencil. Both the question paper and computer sheet must be handed in! 3. For SECTION B all working must be shown! Your answers must be written on the question paper and in the space provided.. No work in pencil or red ink will be marked. 5. If you use correcting fluid ( Tipp- Ex ), you lose the right to question the marking or to indicate work that has not been marked. Outeursreg voorbehou/ Copyright reserved

3 WTW 16 - Lineêre Algebra/Linear Algebra final examination 01 Bladsy/Page 3 SECTION A [multiple choice/veelvuldige keuse] Vraag 1 [1 punt] Question 1 [1 mark] Skryf die komplekse getal ( 3i)(1 + i) 3 + i in stan- Express the complex number ( 3i)(1 + i) 3 + i in daardvorm standard form. (a) i (b) i (c) i (d) i. Vraag [1 punte] Bepaal die komplekse getal z sodat Question [1 marks] Find the complex z such that z + i z = ( + i) + i( i) (a) z = 13 + i 1 (b) z = 1 + i 13 (c) z = 13 i (d) None of these/geen van hierdie. Vraag 3 [1 punte] Indien z = cis(π/5), dan is z 3 gelyk aan Question 3 [1 mark] If z = cis(π/5), then z 3 is equal to (a) 1 cis(3π/5) (b) 6 cis(3π/5) (c) 6 cis( 3π/5) (d) 1 cis(3π/5). Vraag [1 punte] Laat u, v, w drie nie-nul vektore wees. Watter een van die volgende uitdrukkings is ongedefinieer Question [1 mark] Let u, v, w be three non-zero vectors. Which of the following expressions is meaningless (a) v (u w) (b) v u + w u (c) v (u + w) (d) (w u) + v. Vraag 5 [1 punt] As v w = 3 en v w = (1,, ), dan is die hoek tussen die vektore v en w: Question 5 [1 mark] If v w = 3 and v w = (1,, ), then the angle between v and w is : (a) θ = π/3 (b) θ = π/3 (c) θ = 3π/ (d) θ = π/6. Vraag 6 [1 punt] Bepaal α sodat v = (, α, α ) en u = ( 3, α, α) ortogonaal is Question 6 [1 mark] Find α such that v = (, α, α ) and u = ( 3, α, α) are orthogonal (a) α = 5/ (b) α = 1 (c) α = 5/ (d) α = 10 Vraag 7 [1 punte] By watter punt sny die lyn deur (,, 3) en (3, 1, 1) die xy-vlak? Question 7 [1 mark] At what point does the line through the points (,, 3) and (3, 1, 1) intersect the xy-plane? (a) (11/, 1/, 0) (b) (1/, 11/, 0) (c) (,, 0) (d) (11/,, 0). Vraag 8 [1 punte] By watter punt sny die lyne x + 3y = 1 en (x, y) = (1, ) + t( 1, 1)? Question 8 [1 marks] At what point do the lines x + 3y = 1 and (x, y) = (1, ) + t( 1, 1) intersect? (a) (0, 1) (b) (1, 1) (c) ( 1, 0) (d) None of the above. Vraag 9 [1 punte] Die vergelyking van die vlak deur die punt (,, 1) met normaalvektor (, 3, ) is Question 9 [1 mark] The equation of the plane through the point (,, 1) with normal vector (, 3, ) is (a) x 3y z + 1 = 0 (b) x + 3y + z = 1 (c) x + 3y + z = 0 (d) x + 3y + z = 10.

4 WTW 16 - Lineêre Algebra/Linear Algebra Final examination 01 Bladsy/Page Vraag 10 [1 punte] As A n matriks met det A = 5 is, dan is det(3a 1 ) Question 10 [1 marks] If A is a matrix with det A = 5, then det(3a 1 ) is (a) (b) 1 5 (c) 15 (d) Vraag 11 [1 punt] As Question 11 [1 mark] If a b c d e f g h i = 5, bepaal then find a b c d g e h f i g h i (a) 5 (b) 0 (c) 5 (d) None of these/geen van hierdie Vraag 1 [1 punte] As A en B twee inverteerbare matrikse is, dan is (BA) 1 gelyk aan Question 1 [1 marks] If A and B are two invertible matrices, then (BA) 1 is equal to (a) B 1 A 1 (b) A 1 B 1 (c) A 1 + B 1 (d) None of these/geen van hierdie Vraag 13 [1 punte] Gegee dat n wortel is van Question 13 [1 mark] Given that is a root of x 3 + 3x + 3x + = 0, is die komplekse wortels its complex roots are (a) 1 + i 3 and/en 1 i 3 (c) 1 + i 3 and/en 1 + i 3 Vraag 1 [1 punte] As u = (i, 1 i, 1) en w = (0, i, 1 i), dan is (u w) gelyk aan (b) 1 i 3 (d) 1 i 3 and/en 1 i 3 and/en 1 + i 3 Question 1 [1 mark] If u = (i, 1 i, 1) and w = (0, i, 1 i), then (u w) is equal to (a) ( i, i 1, 1) (b) ( i, i+1, 1) (c) (i, i + 1, 1) (d) (i, i + 1, 1) Vraag 15 [1 punte] Die res as x 5 + 3x + i gedeel word deur x + i is Question 15 [1 marks] The remainder when x 5 +3x+i is divided by x+i is (a) 37i (b) 37i (c) 0 (d) i

5 WTW 16 - Lineêre Algebra/Linear Algebra final examination 01 Bladsy/Page 5 SECTION B [Full written answers] Vraag 16a [.5 punte] Bepaal die vergelyking van die vlak deur die punte (1, 0, 1), (0, 1, 1) and ( 1, 1, 0). Question 16a [.5 marks] Find the equation of the plane through the points (1, 0, 1), (0, 1, 1) and ( 1, 1, 0). Vraag 16b [.5 punte] Bepaal die snyding van die vlak in (16a) en die lyn (x, y, z) = (1, 1, ) + t(0, 1, 1). Question 16b [.5 marks] Find the intersection of the plane obtained in (16a) and the line represented as (x, y, z) = (1, 1, ) + t(0, 1, 1).

6 WTW 16 - Lineêre Algebra/Linear Algebra final examination 01 Bladsy/Page 6 Vraag 17a [ marks] Skryf 3 i in poolvorm (modulus argument vorm) Question 17a [ marks] Express 3 i in polar (i.e modulus argument) form Vraag 17b [6 marks] Bepaal alle komplekse getalle z sodat Question 17b [6 marks] Find all complex numbers z such that z = 3 i, en toon al jou oplossings in die komplekse vlak aan. Jy mag jou antwoorde in poolvorm los. and represent all your solutions in the complex plane. You may leave your answer in polar form.

7 WTW 16 - Lineêre Algebra/Linear Algebra final examination 01 Bladsy/Page 7 Vraag 17c [3+1 marks] Question 17c [3+1 marks] Toon ( ) aan dat die komplekse getalle z wat Show 1 Re = 1 ( ) that the complex numbers z satisfying 1 bevredig op n sirkel lê. Teken die Re = 1 form a circle. Draw the circle, indicating its centre and z z sirkel en dui sy middelpunt en radius aan. radius

8 WTW 16 - Lineêre Algebra/Linear Algebra final examination 01 Bladsy/Page 8 Vraag 18(a) [ marks] Bepaal die determinant van A = Question 18(a) [ marks] Find the determinant of, en dui aan of A inverteerbaar is of nie. Regverdig jou antwoord. and indicate whether A is invertible or not. Justify your answer. Vraag 18(b) [5 marks] Bepaal die inverse van A deur Gauss reduksie te gebruik. Dui al jou berekeninge aan deur elke Gauss reduksie stap te beskryf. Question 18(b) [5 marks] Find the inverse of A using Gauss reduction. Show your working, and label each Gauss reduction step clearly.

9 WTW 16 - Lineêre Algebra/Linear Algebra final examination 01 Bladsy/Page 9 Vraag 18(c) [3 marks] Vir A gegee in (18a), los op vir X in die stelsel vergelykings XA = [ Question 18(c) [3 marks] For A given in (18a), solve for X in the system of equations ] Wenk: Jy mag die inverse in (18b) gebruik. Hint: you may use the inverse obtained in (18b) Vraag 19a [ marks] Jy het Gauss reduksie toegepas op die aangevulde matriks en jy kry die volgende resultaat Question 19a [ marks] You have performed Gauss-reduction on the augmented matrix and obtained the following result k k k k 1 Vir watter waardes van k het die stelsel vergelykings met veranderlikes (x, y, z) geen oplossings nie? (Verduidelik hoekom) For which values of k does this system of linear equations in variables (x, y, z) have no solution? (explain why) Vraag 19b [ marks] Vir watter waardes van k het die stelsel vergelykings oneindig baie oplossings. Vir hierdie waarde van k, gee die oplossing in vektorvorm en toon al jou berekeninge. Question 19b [ marks] For which value of k does the system have infinitely many solutions?. For this value of k, give the solutions in vector form and show all working.

10 WTW 16 - Lineêre Algebra/Linear Algebra final examination 01 Bladsy/Page 10 Vraag 0 [3 marks] Gestel dat A en B n n matrikse is en laat I die n n identiteitsmatriks wees. As B 1 en (AB + I) 1 beide bestaan, toon aan dat Question 0 [3 marks] (A + B 1 ) 1 = B(AB + I) 1. Suppose that A and B are n n matrices and let I be the n n identity matrix. If B 1 and (AB + I) 1 both exist show that

11 WTW 16 - Lineêre Algebra/Linear Algebra final examination 01 Bladsy/Page 11 ROUGH WORKING einde/end

UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS

VAN/SURNAME: UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS VOORNAME/FIRST NAMES: WTW 162 DYNAMICAL PROCESSES EKSAMEN