<p> 2012 Mu Alpha Theta National Convention Alpha Matrices and Vectors</p><p>For all questions, answer E. "NOTA" means none of the above answers is correct. Let In denote the n n identity matrix and let 0m n denote the m n matrix of all zeros.</p><p>轾3- 1 0 轾1- 3 1 A = 犏2 7 4 B = 犏0 2 0 For the questions 1 and 2, let 犏 and 犏 . 臌犏-2 1 1 臌犏-1 5 4</p><p>1. Evaluate A+ B . 轾4- 4 1 轾6- 2 - 8 犏2 9 4 犏-15 14 49 A. 犏 B. 犏 臌犏-3 6 5 臌犏-4 2 11 轾3 3 0 轾3- 11 3 犏0 14 0 犏-2 28 18 C. 犏 D. 犏 E. NOTA 臌犏2 5 4 臌犏-3 13 2</p><p>2. Evaluate A B . 轾4- 4 1 轾6- 2 - 8 犏2 9 4 犏-15 14 49 A. 犏 B. 犏 臌犏-3 6 5 臌犏-4 2 11 轾3 3 0 轾3- 11 3 犏0 14 0 犏-2 28 18 C. 犏 D. 犏 E. NOTA 臌犏2 5 4 臌犏-3 13 2</p><p>轾2 轾-4 x = 犏2 Ax = 犏4 3. Let 犏 and let A be a 3 3 matrix such that 犏 . What is the sum of the entries 臌犏2 臌犏16 of A? A. 2 B. 4 C. 8 D. 16 E. NOTA</p><p>4. Which of the following vectors has the same direction as 1,8,- 4 and magnitude 2? 1 8 4 2 16 8 1 8 4 2 16 8 A. , ,- B. , ,- C. , , - D. , , - E. NOTA 9 9 9 9 9 9 3 3 3 3 3 3</p><p>5. Jeff, Brian, and Alex decide to go to an organic grocery store to buy exotic fruit. Jeff buys two apples, one orange, and three bananas for $19. Brian buys one apple, three oranges, and a banana for $10. Alex buys four apples and two oranges for $14. How much does an apple cost? A. $2.00 B. $2.50 C. $3.00 D. $3.50 E. NOTA</p><p>Page 1 of 5 Alpha Matrices and Vectors M A  National Convention 2012 </p><p>1- 1 0 1 2 3 0 3 6. Evaluate -4 1 0 1 7 9 1- 5 A. -28 B. -4 C. 4 D. 28 E. NOTA</p><p>轾1- 1 7. Let A = 犏 and let B be a matrix with real entries such that AB= BA. What is the 臌1 1 smallest possible value of B . 1 A. -1 B. 0 C. D. 1 E. NOTA 2</p><p>8. Find the area of the parallelogram in the xy-plane with vertices (2,5) , (5,7) , (9,13) , and (6,11) . A. 5 B. 10 C. 15 D. 20 E. NOTA</p><p>9. Let A and B be n n matrices such that A = 2 and B = 3 . What is AT B T ? 2 3 A. B. 1 C. D. 5 E. NOTA 3 2</p><p>轾x -1 3 犏2 0 x 10. For what value of x is the matrix 犏 not invertible? 臌犏4- 2 7 A. -1 B. 0 C. 1 D. 2 E. NOTA</p><p>11. Which of the following is a unit vector normal to the plane 2x- 3 y + 6 z = 13 ? 3 2 3 6 A. 6,- 3,2 B. 2,- 3,6 C. 1,- ,3 D. ,- , E. NOTA 2 7 7 7</p><p>12. The matrix A transforms the points (0,0) , (2,0) , and (1,1) to the points (0,0) , (0,4) , and (3,3) , respectively. What is det A ? A. 1 B. 2 C. 4 D. 6 E. NOTA</p><p>2 3 13. Let A be an n n matrix such that A-3 A + 2 In = 0 n n . What is A written in terms of A and In ?</p><p>A. 3A- 2 In B. 6A+ 4 In C. 7A- 6 In D. 9A+ 4 In E. NOTA</p><p>14. Which of the following is the result of rotating the vector 2 3,1 counter-clockwise 60 ? </p><p>Page 2 of 5 Alpha Matrices and Vectors M A  National Convention 2012 3 10 3 7 A. , B. , C. 1,2 3 D. 3, 10 E. NOTA 3 3 2 2</p><p>For questions 15, 16, and 17 let u =3, - 2,5 and v =1,0, - 2 .</p><p>15. Evaluate 3u- v . A. 0,- 2,11 B. 6,- 6,21 C. 8,- 6,17 D. 12,- 6,9 E. NOTA</p><p>16. Evaluate u v . A. -7 B. 0 C. 2 D. 13 E. NOTA</p><p>17. Evaluate u v . A. 2,- 2,7 B. 3,0,- 10 C. 4,- 2,3 D. 4,11,2 E. NOTA</p><p>轾1 2 ⋯ n 犏 犏2 4⋯ 2n 18. Let A be the n n matrix A = , where n 4 . What is the rank of A? 犏⋮ ⋮ ⋱ ⋮ 犏 犏 2 臌n2 n⋯ n A. 1 B. 2 C. n -1 D. n E. NOTA</p><p>19. Let A be an invertible 3 3 matrix and let B be a 3 5 matrix. What are the dimensions of BT A-1 B ? A. 3 3 B. 3 5 C. 5 3 D. 5 5 E. NOTA</p><p>轾2 1 轾 7- 6 20. Let A be a 2 2 matrix such that 犏A = 犏 . What is the sum of the entries of 臌4 3 臌 19- 12 A? A. -7 B. 3 C. 10 D. 15 E. NOTA</p><p>21. What is the measure of the smaller angle in degrees between the vectors (3,1) and (2,4) ? A. 45 B. 60 C. 90 D. 135 E. NOTA</p><p>For questions 22 and 23, let A be the 9 9 matrix where every entry is 3.</p><p>22. What is A ? A. 0 B. 3 C. 93 D. 39 E. NOTA</p><p>23. What is the trace of A? A. 0 B. 3 C. 9 D. 27 E. NOTA</p><p>24. Which of the following matrices reflects the point ( x, y) across the line y= x ?</p><p>Page 3 of 5 Alpha Matrices and Vectors M A  National Convention 2012 轾1 0 轾0 0 轾1 0 轾0 1 A. 犏 B. 犏 C. 犏 D. 犏 E. NOTA 臌0 0 臌0 1 臌1 0 臌1 0</p><p>Page 4 of 5 Alpha Matrices and Vectors M A  National Convention 2012 25. How many of the following statements are true concerning two invertible n n matrices A and B?</p><p>2 I. ( A+ B) = A2 +2 AB + B 2 2 II. ( AB) = A2 B 2 -1 III. ( A+ B) = A-1 + B - 1 -1 IV. ( AB) = A-1 B - 1 A. 1 B. 2 C. 3 D. 4 E. NOTA</p><p>26. How many distinct solutions ( x, y) does the system of equations 3x+ 2 y = - 5 , -2x + y = 8 , and 4x+ 5 y = - 2 have? A. 0 B. 1 C. 2 D. Infinitely many E. NOTA</p><p>27. Let u and v be two vectors in the xy-plane with initial points at the origin and let the terminal point of u be in the first quadrant. If u� v - 3 , then in which of the following can be true?</p><p>I. The terminal point of v is in the first quadrant. II. The terminal point of v is in the second quadrant. III. The terminal point of v is in the third quadrant. IV. The terminal point of v is in the fourth quadrant.</p><p>A. I and III B. II and IV C. II, III, and IV D. I, II, III, and IV E. NOTA</p><p>轾2 2 28. For which of the following values of l is there a non-zero vector x such that 犏 x= l x 臌1 3 ? A. 0 B. 1 C. 2 D. 3 E. NOTA</p><p>轾0 1 2 A = 犏0 0 1 2012 29. Let 犏 . What is A ? 臌犏0 0 0 轾0 1 2 轾0 1 22012 犏 犏0 0 1 0 0 1 A. 犏 B. 犏 犏 臌犏0 0 0 臌0 0 0 轾0 1 0 轾0 2012 2 犏0 0 1 犏0 0 2012 C. 犏 D. 犏 E. NOTA 臌犏0 0 0 臌犏0 0 0</p><p>30. Which of the following physical quantities can not be represented by a vector? A. Acceleration B. Density C. Force D. Velocity E. NOTA</p><p>Page 5 of 5</p>