Mu Alpha Theta Convention 2004

Theta Matrices and Determinants Test

For all questions, answer E. "NOTA" means none of the above answers is correct.

If , then evaluate .

A. 6B. 12C. 24D. 48E. NOTA

Given: and a = sum of the elements in row 1, b = product of the elements in column 2, and c = determinant of matrix X. Find:

A. 7B. C. -7D. E. NOTA

3. If is a matrix and , then evaluate

A.5B. -5C. D. E. NOTA

4. A is a square matrix, is the identity matrix of the same order as and . Which of the following is not

necessarily equal to 3?

A.B. C. D. E. NOTA

5. Find the sum of the entries of

A. 6B. C. D. E. NOTA

6. Given: and . What are the dimensions of ?

A.B. C. is not definedD. E. NOTA

Mu Alpha Theta Convention 2004

Theta Matrices and Determinants Test

For all questions, answer E. "NOTA" means none of the above answers is correct.

7. What is the value of ?

A.9B. 6C. 0D. 18E. NOTA

8. If matrix , , has an inverse, then

A.B. C. D. 0E. NOTA

9. Solve for if

A.-2B. 2C. -3D. 3E. NOTA

10. Find the sum of all values of such that the system has no solution.

A.B. 0C. 21 D. 27E. NOTA

11. . Solve for if is a matrix.

A.228B. 56C. 0D. -56E. NOTA

12. Find , given

A. B. C. D. E. NOTA

13. Let What is given ?

A.1B. 4C. D. 16E. NOTA

14. Given: . If , what is

A.1B. 2C. -21D. -52E. NOTA

Mu Alpha Theta Convention 2004

Theta Matrices and Determinants Test

For all questions, answer E. "NOTA" means none of the above answers is correct.

Given Z = . If , then find .

A.6B. 3C. -6D. -18E. NOTA

16. Find the area of a triangle with vertices .

A.5B. 5.5C. 6D. 11E. NOTA

17. Which of the following is/are not characteristics of matrices in row echelon form?

All nonzero rows are preceded by zero rows (if both are present) .
The last (right) nonzero element of each row is 1
When the first nonzero element of a row appears in column C, then all elements in column C in succeeding rows are zero

A.I, II, IIIB. II, IIIC. II onlyD. III onlyE. NOTA

18. Given , which of the following are true?

The determinant of is positive.
The product yields a matrix.
The product yields a matrix.
The determinant of the inverse of equals the determinant of .

A.I onlyB. I, II and IVC. III onlyD. I and IIE. NOTA

19. If the determinant of and the determinant of , find .

A. B. C. D. E. NOTA

20. If , find the sum of all possible determinants of

A.10B. 11C. 12D. 13E. NOTA

21. Which matrix satisfies the equation:

A.B. C. D. E. NOTA

22. There is a 50% chance of rain tomorrow if it is raining today. If it is sunny today, there is only a 30% chance of rain tomorrow. Given that it is sunny today, what is the chance of rain this time next week?

A.30%B. 37.5%C. 50%D. 62.5%E. NOTA

Mu Alpha Theta Convention 2004

Theta Matrices and Determinants Test

For all questions, answer E. "NOTA" means none of the above answers is correct.

23. Given the three points are collinear, find the value of .

A. -26B. -16C. 16D. 26E. NOTA

Which of the following matrices commute?

I. II. III.

A.III onlyB. I and IIC. I and IIID. I, II and IIIE. NOTA

25. If are matrices and is an matrix, then which of the following are not true?

A. B. C.

D.E. NOTA

26. If and the determinant of is 17, solve for .

A. -1B. 1C. 3D. 4E. NOTA

27. Given ,

A. B. C. D. E. NOTA

28. Determine the cofactor of the element in row 2, column 3 of the matrix

A.0B. 18C. -18D. -12E. NOTA

29. What is the element in the third row, first column of the adjoint of the matrix ?

A. -2B. 4C. 10D. -18E. NOTA

30. What is the sum of the eigenvalues of the matrix ?

A.2B. 3C. 9D. 12E. NOTA

Mu Alpha Theta Convention 2004

Theta Matrices and Determinants Test

Tiebreakers

1. What values of x will satisfy the inequality ?

2. Joel was shown the solution of and for a system of equations using Cramer's Rule. From this information, he was able to set up the solution for . What value did he get for ?

What is the determinant of if ?

Mu Alpha Theta National Convention 2004

Theta Matrices & Determinants

Answers

# /

Answer

/ # /

Answer

1 / D / 18 / D
2 / D / 19 / D
3 / C / 20 / B
4 / C / 21 / A
5 / C / 22 / B
6 / B / 23 / C
7 / A / 24 /

Thrown out

8 / A / 25 /

Thrown out

9 / D / 26 / E
10 / B / 27 / A
11 / E / 28 / B
12 / D / 29 / C
13 / C / 30 / A
14 / D / TB1 / -2/3<x<0
15 / E / TB2 / -2
16 / B / TB3 / 8-30x
17 / E

Solutions:

1. D 8 (6) = 48

2. D a = 11, b = 133, c = -15

3. C | AT | = | A | = 5;

4. C

5. C

6. B A = 3 x 2 B = 2 x 3 AB = 3 x 3

7. A 6 + 3 = 9

8. A

9. D

Solving the system, x = 3, y = -1

10. B

sum of the roots of k are

11. E

| X | = 4.8

12. D 2x + 3y = 7 8x + 4y = 1

13. C ab-cb =

14. D 4 - 2c = 2 8b – 42 = -26

c = 1 b = 2

15. E e = 0 so the answer = 0

16. B 1 -1 .5 ( 4 + 6 + 0 + 3 – 0 – 2)

3 4 .5 (11)

0 2 5.5

1 -1

17. E

18. D | A | = 13 AB is a 2 x 3

BA is undefined

I and II are true

19. D xy – 24 = -20 x2 + 2xy + y2 = 49

xy = 4 (x + y)2 = 49

x + y =

x3 + y3 = (x + y)(x2 – xy + y2)

x3 + y3 = ()(49 – 2(4) – 4) =

20. B 2x – 15 28x – 5x – 21x + 18

4x – 7

x3 – 8x2 + 17x – 10 = 0 x = 1, 2, 5

4(1) – 7 = -3; 4(2) – 7 = 1;

4(5) – 7 = 13 -3 + 1 + 13 = 11

21. A use a calculator

22. B

23. C 3 points are collinear if

4x + 10 + 3y – 20 – xy – 6 = 0

4x – xy + 3y = 16

24. thrown out

25. E all are true (question thrown out)

26. E 2x2 + 18 + 40 – 4x2 – 15 – 24 = 17

-2x2 = -2 x =

27. A

28. B

29. C

30. A sum of the eigenvalues = sum of the

elements in the main diagonal

0 + 2 = 2

TB 1

-20 – 2x(3x + 2) > -20

-2x(3x + 2) > 0

TB 2 -2

TB 3 8 – 30x

3(0 + 1 – 5 – 0 – 0 – 5x) + 1(0 + 3 + 15 + 2 – 15x)

3(-4 – 5x) + (20 – 15x)

8 – 30x