Chapter 2Exam

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Solve the system of equations.1) x1 - x2 + 3x3 = -8

2x1 + x3 = 0x1 + 5x2 + x3 = 40

A) (0, 8, 0) B) (0, -8, -8) C) (8, 8, 0) D) (-8, 0, 0)Answer: A

2) 5x1 + 2x2 + x3 = -112x1 - 3x2 - x3 = 177x1 + x2 + 2x3 = -4

A) (0, -6, 1) B) (0, 6, -1) C) (3, 0, -4) D) (-3, 0, 4)Answer: A

Determine whether the system is consistent.3) x1 + x2 + x3 = 7

x1 - x2 + 2x3 = 75x1 + x2 + x3 = 11

A) No B) YesAnswer: B

4) 4x1 - x2 + 3x3 = 122x1 + 9x3 = -5x1 + 4x2 + 6x3 = -32

A) No B) YesAnswer: B

5) 2x1 + x2 = 0x1 - 3x2 + x3 = 03x1 + x2 - x3 = 0

A) Yes B) NoAnswer: A

6) x1 - x2 + 4x3 = -15-5x1 + 5x2 - 20x3 = -4x1 + 3x2 + x3 = -1

A) No B) YesAnswer: A

1

Determine whether the matrix is in echelon form, reduced echelon form, or neither.

7) 1 3 5 -7 0 1 -4 1 0 0 1 8

A) Neither B) Reduced echelon form C) Echelon formAnswer: C

8) 1 4 5 -7 0 1 -4 8 0 6 1 7

A) Reduced echelon form B) Echelon form C) NeitherAnswer: C

9) 1 4 5 -72 1 -4 5 0 2 1 2

A) Reduced echelon form B) Echelon form C) NeitherAnswer: C

10) 1 0 0 -75 1 0 -3 0 4 1 5

A) Reduced echelon form B) Echelon form C) NeitherAnswer: C

11) 1 6 7 -7 0 1 -4 -9 0 0 0 0

A) Reduced echelon form B) Neither C) Echelon formAnswer: C

12)

1 0 -4 20 1 -5 -50 0 0 00 0 0 0

A) Neither B) Reduced echelon form C) Echelon formAnswer: B

13)

1 -2 -4 40 0 4 -30 0 0 -20 0 0 0

A) Reduced echelon form B) Neither C) Echelon formAnswer: C

2

Use the row reduction algorithm to transform the matrix into echelon form or reduced echelon form as indicated.14) Find the echelon form of the given matrix.

1 4 -2 3-3 -11 9 -5-2 -3 3 -1

A)1 4 -2 30 1 3 40 0 -16 0

B)1 4 -2 30 1 3 40 0 -6 0

C)1 4 -2 30 1 3 40 0 -16 -15

D)1 4 -2 30 1 3 40 5 -1 5

Answer: C

15) Find the reduced echelon form of the given matrix. 1 4 -5 1 2 2 5 -4 -1 4-3 -9 9 2 2

A) 1 0 0 0 14 0 1 0 0 -4 0 0 0 1 4

B) 1 4 -5 1 2 0 1 -2 1 0 0 0 0 1 4

C) 1 4 -5 0 -2 0 1 -2 0 -4 0 0 0 1 4

D) 1 0 3 0 14 0 1 -2 0 -4 0 0 0 1 4

Answer: D

The augmented matrix is given for a system of equations. If the system is consistent, find the general solution. Otherwisestate that there is no solution.

16) 1 -5 -4 0 0 2

A) (-4, 2) B) No solution C) x1 = -4 + 5x2x2 = 2x3 is free

D) x1 = -4 + 5x2x2 is free

Answer: B

17) 1 2 -3 -19 0 1 4 4 0 0 0 1

A) x1 = -27 + 11x3x2 = 4 - 4x3x3 is free

B) x1 = -19 - 2x2 + 3x3x2 is freex3 is free

C) x1 = -27 + 11x3x2 = 4 - 4x3x3 = 1

D) No solution

Answer: D

3

18) 1 2 -3 8 0 1 4 -3 0 0 0 0

A) x1 = 14 + 11x3x2 = -3 - 4x3x3 is free

B) x1 = 14 + 11x3x2 = -3 - 4x3x3 = 0

C) x1 = 8 - 2x2 + 3x3x2 is freex3 is free

D) x1 = 8 -2x2 + 3x3x2 = -3 - 4x3x3 is free

Answer: A

19) 1 0 6 5 0 1 -2 -9 0 0 0 0

A) x1 = 5 - 6x3x2 is free

x3 =92

+12

x2

B) x1 = 5 - 6x3x2 = -9 + 2x3x3 = 0

C) x1 = 5 - 6x3x2 = -9 + 2x3x3 is free

D) No solution

Answer: C

20) 1 4 -2 -3 1 0 0 1 3 5-1 -4 -2 -9 -21

A) x1 = 11 - 4x2 - 3x3x2 = 5 - 3x3x3 is free

B) x1 = - 4x2 +2x3 + 3 x4 + 1x2 is freex3 = 5 - 3x4x4 is free

C) x1 = 11 - 4x2 - 3x4x2 is freex3 = 5 - 3x4x4 = 0

D) x1 = 11 - 4x2 - 3x4x2 is freex3 = 5 - 3x4x4 is free

Answer: D

4

21)1 4 6 -1 2 80 0 0 -4 3 40 0 0 0 -2 8

A) x1 = -4x2 - 6x3 + 12x2 is freex3 is freex4 = -4x5 = -4

B) x1 = -4x2 - 6x3 + 12x2 is freex3 = -4

x4 =34 x5 - 1

x5 = -4

C) x1 = -4x2 - 6x3 + x4 - 2x5 + 8x2 is freex3 is free

x4 =34

x5 - 1

x5 = -4

D) No solution

Answer: A

5