Alpha Matrices and Vectors MAQ National Convention 2012

2012 Mu Alpha Theta National Convention Alpha Matrices and Vectors

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.

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

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

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

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

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

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

of 5 Alpha Matrices and Vectors M A  National Convention 2012

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

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

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

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

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

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

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

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 ?

A. 3A- 2 In B. 6A+ 4 In C. 7A- 6 In D. 9A+ 4 In E. NOTA

14. Which of the following is the result of rotating the vector 2 3,1 counter-clockwise 60 ?

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

For questions 15, 16, and 17 let u =3, - 2,5 and v =1,0, - 2 .

15. Evaluate 3u- v . A. 0,- 2,11 B. 6,- 6,21 C. 8,- 6,17 D. 12,- 6,9 E. NOTA

16. Evaluate u v . A. -7 B. 0 C. 2 D. 13 E. NOTA

17. Evaluate u v . A. 2,- 2,7 B. 3,0,- 10 C. 4,- 2,3 D. 4,11,2 E. NOTA

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

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

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

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

For questions 22 and 23, let A be the 9 9 matrix where every entry is 3.

22. What is A ? A. 0 B. 3 C. 93 D. 39 E. NOTA

23. What is the trace of A? A. 0 B. 3 C. 9 D. 27 E. NOTA

24. Which of the following matrices reflects the point ( x, y) across the line y= x ?

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

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?

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

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

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?

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.

A. I and III B. II and IV C. II, III, and IV D. I, II, III, and IV E. NOTA

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

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

30. Which of the following physical quantities can not be represented by a vector? A. Acceleration B. Density C. Force D. Velocity E. NOTA