Chapter 2 Exam 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 = 0 x1 + 5x2 + x3 = 40
A) (0, 8, 0)
B) (0, -8, -8)
C) (8, 8, 0)
D) (-8, 0, 0)
B) (0, 6, -1)
C) (3, 0, -4)
D) (-3, 0, 4)
Answer: A 2) 5x1 + 2x2 + x3 = -11 2x1 - 3x2 - x3 = 17 7x1 + x 2 + 2x3 = -4
A) (0, -6, 1)
Answer: A Determine whether the system is consistent. 3) x1 + x2 + x3 = 7 x1 - x2 + 2x3 = 7 5x1 + x 2 + x3 = 11 A) No
B) Yes
Answer: B 4) 4x1 - x 2 + 3x3 = 12 2x1 + 9x3 = -5
x1 + 4x 2 + 6x3 = -32
A) No
B) Yes
Answer: B 5) 2x1 + x 2 = 0 x1 - 3x 2 + x3 = 0 3x1 + x 2 - x3 = 0
A) Yes
B) No
Answer: A 6) x1 - x2 + 4x3 = -15
-5x1 + 5x2 - 20x3 = -4 x1 + 3x2
A) No
+ x3 = -1
B) Yes
Answer: A
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Determine whether the matrix is in echelon form, reduced echelon form, or neither. 1 3 5 -7 7) 0 1 -4 1 0 0 1 8 A) Neither
B) Reduced echelon form
C) Echelon form
B) Echelon form
C) Neither
B) Echelon form
C) Neither
B) Echelon form
C) Neither
B) Neither
C) Echelon form
B) Reduced echelon form
C) Echelon form
B) Neither
C) Echelon form
Answer: C 8)
1 4 5 -7 0 1 -4 8 0 6 1 7
A) Reduced echelon form Answer: C 1 4 5 -7 9) 2 1 -4 5 0 2 1 2
A) Reduced echelon form Answer: C 1 0 0 -7 10) 5 1 0 -3 0 4 1 5
A) Reduced echelon form Answer: C 11)
1 6 7 -7 0 1 -4 -9 0 0 0 0
A) Reduced echelon form Answer: C 1 12) 0 0 0
0 -4 2 1 -5 -5 0 0 0 0 0 0
A) Neither Answer: B 1 -2 -4 4 13) 0 0 4 -3 0 0 0 -2 0 0 0 0
A) Reduced echelon form Answer: C
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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) B) C) D) 1 4 -2 3 1 4 -2 3 1 4 -2 3 1 4 -2 3 0 1 3 4 0 1 3 4 0 1 3 4 0 1 3 4 0 0 -16 0 0 0 -6 0 0 0 -16 -15 0 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 C) 1 4 -5 0 -2 0 1 -2 0 -4 0 0 0 1 4
B)
D)
1 0 0
4 -5 1 2 1 -2 1 0 0 0 1 4
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. Otherwise state that there is no solution. 16) 1 -5 -4 0 0 2
A) (-4, 2)
C) x1 = -4 + 5x2
B) No solution
x2 = 2
x3 is free
Answer: B 17)
1 2 -3 -19 0 1 4 4 0 0 0 1
A) x1 = -27 + 11x3
B) x1 = -19 - 2x2 + 3x3
x2 = 4 - 4x3 x3 is free
x2 is free x3 is free
C) x1 = -27 + 11x3
D) No solution
x2 = 4 - 4x3 x3 = 1
Answer: D
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D) x1 = -4 + 5x2 x2 is free
18)
1 2 -3 8 0 1 4 -3 0 0 0 0
A) x1 = 14 + 11x3 x2 = -3 - 4x3 x3 is free
B) x1 = 14 + 11x3
C) x1 = 8 - 2x2 + 3x3
x2 = -3 - 4x3
x2 is free
x3 = 0
x3 is free
D) x1 = 8 -2x2 + 3x3 x2 = -3 - 4x3 x3 is free
Answer: A 19)
1 0 6 5 0 1 -2 -9 0 0 0 0
A) x1 = 5 - 6x3 x2 is free
9 1 x3 = + x 2 2 2
B) x1 = 5 - 6x3 x2 = -9 + 2x3
C) x1 = 5 - 6x3
x2 = -9 + 2x3
x3 = 0
x3 is free
Answer: C 20)
1 4 -2 -3 1 0 0 1 3 5 -1 -4 -2 -9 -21
A) x1 = 11 - 4x2 - 3x3
B) x1 = - 4x2 +2x3 + 3 x4 + 1 x2 is free
x2 = 5 - 3x3
x3 = 5 - 3x4
x3 is free
x4 is free
C) x1 = 11 - 4x2 - 3x4
D) x1 = 11 - 4x2 - 3x4
x2 is free
x2 is free
x3 = 5 - 3x4
x3 = 5 - 3x4
x4 = 0
x4 is free
Answer: D
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D) No solution
1 4 6 -1 2 8 21) 0 0 0 -4 3 4 0 0 0 0 -2 8
A) x1 = -4x2 - 6x3 + 12 x2 is free
B) x1 = -4x2 - 6x3 + 12 x2 is free
x3 is free
x3 = -4
x4 = -4
x4 =
x5 = -4
x5 = -4
C) x1 = -4x2 - 6x3 + x4 - 2x5 + 8 x2 is free
D) No solution
x3 is free x4 =
3 x -1 4 5
3 x -1 4 5
x5 = -4
Answer: A
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