Precalculus: Chapter 8 Review Team #: ______

(NON-CALCULATOR) Period: ______

Multiple Choice

Identify the choice that best completes the statement or answers the question.

Write the letter for the correct answer in the blank at the right of each question.

____ 1. What is the augmented matrix for the given system?

a. / / c. /
b. / / d. /

____ 2. Which matrix is not in row-echelon form?

a. / / c. /
b. / / d. /

____ 3. Choose the phrase that best describes the matrix.

a. / / c. /
b. / / d. / none of the above

____ 4. Solve the system of equations using a matrix and Gaussian elimination.

a. / (4, 2) / c. / (4, -2)
b. / (-4, -2) / d. / (-4, 2)

____ 5. Solve the following system of equations using an inverse matrix.

4x + 5y = –21

–2x – 4y = 6

a. / / c. /
b. / no solution / d. /

Precalculus: Chapter 8 Review Team #: ______

(NON-CALCULATOR) Period: ______

____ 6. What is the determinant of ?

a. / -8 / c. / 12
b. / 8 / d. / 20

____ 7. Find DE if and .

a. / / c. /
b. / / d. /

____ 8. Find the inverse of , if it exists.

a. / does not exist / c. /
b. / / d. /

____ 9. What is B if and .

a. / / c. /
b. / / d. /

Precalculus: Chapter 8 Review Team #: ______

(CALCULATOR) Period: ______

____ 10. Write a matrix equation for the given systems of equations.

a. / / c. /
b. / / d. /

____ 11. Solve the following system of equations using an inverse matrix.

a. / (1, 0, -2) / c. / (-1, 0, 2)
b. / (-1, 0, -2) / d. / (1, 0, 2)

____ 12. Solve the system of equations.

10x + 24y + 2z = –18

–2x – 7y + 4z = 6

–14x – 48y + 26z = 42

a. / x = –8, y = 2, z = 7 / c. / infinite solutions
b. / x = 7, y = 6, z = –10 / d. / no solution

____ 13. What is the determinant of ?

a. / -151 / c. / 141
b. / -141 / d. / 151

____ 14. FOOD The table shows several boxes of assorted candy available at a candy shop. What is the price per pound for each candy?

a. / ($0.85, $0.75, $0.80) / c. / ($0.80, $0.75, $0.85)
b. / ($0.75, $0.80, $0.85) / d. / ($0.75, $0.85, $0.80)

Precalculus: Chapter 8 Review Team #: ______

(CALCULATOR) Period: ______

____ 15. Solve the system of equations using a matrix and Gauss-Jordan elimination.

2x – 3y + z = –14

14x – 18y + 12z = –30

–15x + 21y – 9z = 81

a. / x = –3, y = 6, and z = 10 / c. / x = –5, y = –3, and z = –1
b. / x = 5, y = 8, and z = 0 / d. / no solution

____ 16. Solve the system of equations.

2x – 2y + 6z – 26w = 30

–2x + y – 6z + 21w = –33

3x – 3y + 6z – 21w = 21

a. / (–6 + 8w, 3 – 3w, 9 – w, w) / c. / (6, 113, 6, –8)
b. / (–1 + 5w, 6 + 4w, –7 – 8w, w) / d. / (–6 – 10w, 3 – 5w, 8 + 6w, w)

____ 17. Determine whether and are inverse matrices.

a. / Yes / b. / No

____ 18. Solve the matrix equation by using inverse matrices.

a. / (, ) / c. / (–2, )
b. / (, 5) / d. / (–2, 5)

____ 19. Use an inverse matrix to solve the system of equations, if possible.

5x + 4y + z = –73

3x – 6y + 3z = 45

–4x + 8y – z = –33

a. / / c. / no solution
b. / / d. /

Precalculus: Chapter 8 Review Team #: ______

Period: ______

Short Answer

20. If , find .

21. If A and B are inverse 2 x 2 matrices, what matrix represents the product of A and B?

22. What is the inverse of matrix A, if ?

23. Find the determinant of using minor and cofactors.

24. Find the determinant of using minor and cofactors.

25. Find the inverse of , if it exists. Use both calculator and non-calculator approaches.

Precalculus: Chapter 8 Test Review

Answer Section

MULTIPLE CHOICE

1. ANS: C PTS: 1

2. ANS: D PTS: 1

3. ANS: C

Feedback
A / There is a column of constant terms.
B /
C / Correct!
D /

PTS: 1 DIF: Average REF: Lesson 6-1

OBJ: 6-1.1 Solve systems of linear equations using matrices and Gaussian elimination.

NAT: 2 STA: 8.D.5 TOP: Multivariable Linear Systems and Row Operations

NOT: Example 3: Identify an Augmented Matrix in Row-Echelon Form

4. ANS: D PTS: 1

5. ANS: C

Feedback
A / Substitute x = –9 and y = –1 back into each equation in the system.
B / The system has a unique solution.
C / Correct!
D / Substitute x = –4 and y = –1 back into each equation in the system.

PTS: 1 DIF: Average REF: Lesson 6-3

OBJ: 6-3.2 Solve systems of linear equations using Cramer's Rule.

NAT: 2 STA: 8.D.5 TOP: Solving Linear Systems Using Inverses and Cramer's Rule

KEY: Matrices | Systems of Linear Equations | Cramer's Rule

NOT: Example 3: Use Cramer's Rule to Solve a 2x2 System

6. ANS: B PTS: 1

7. ANS: A PTS: 1

8. ANS: B PTS: 1

9. ANS: B PTS: 1

10. ANS: D PTS: 1 DIF: Average REF: Lesson 6-1

OBJ: 6-1.1 Solve systems of linear equations using matrices and Gaussian elimination.

NAT: 2 STA: 8.D.5 TOP: Multivariable Linear Systems and Row Operations

KEY: Matrix Equations | Systems of Equations

NOT: Example 2: Write an Augmented Matrix

11. ANS: C PTS: 1

12. ANS: D

Feedback
A / Check the steps of the Gaussian elimination.
B / Check the steps of the Gaussian elimination.
C / Check the steps of the Gaussian elimination.
D / Correct!

PTS: 1 DIF: Advanced REF: Lesson 6-1

OBJ: 6-1.2 Solve systems of linear equations using matrices and Gauss-Jordan elimination.

NAT: 2 STA: 8.D.5 TOP: Multivariable Linear Systems and Row Operations

NOT: Example 6: No Solution and Infinitely Many Solutions

13. ANS: C PTS: 1

14. ANS: A PTS: 1

15. ANS: A

Feedback
A / Correct!
B / Check the steps of the Gauss-Jordan elimination.
C / Check the steps of the Gauss-Jordan elimination.
D / Check the steps of the Gauss-Jordan elimination.

PTS: 1 DIF: Advanced REF: Lesson 6-1

OBJ: 6-1.2 Solve systems of linear equations using matrices and Gauss-Jordan elimination.

NAT: 2 STA: 8.D.5 TOP: Multivariable Linear Systems and Row Operations

NOT: Example 5: Use Gauss-Jordan Elimination

16. ANS: D

Feedback
A / Check the steps of the Gauss-Jordan elimination.
B / Check the steps of the Gauss-Jordan elimination.
C / Check the steps of the Gauss-Jordan elimination.
D / Correct!

PTS: 1 DIF: Advanced REF: Lesson 6-1

OBJ: 6-1.2 Solve systems of linear equations using matrices and Gauss-Jordan elimination.

NAT: 2 STA: 8.D.5 TOP: Multivariable Linear Systems and Row Operations

NOT: Example 5: Use Gauss-Jordan Elimination

17. ANS: B PTS: 1 DIF: Average REF: Lesson 6-2

OBJ: 6-2.2 Find determinants and inverses of 2x2 and 3x3 matrices.

NAT: 1 STA: 8.C.4b TOP: Matrix Multiplication, Inverses, and Determinants

KEY: Matrices | Inverses of Matrices NOT: Example 4: Verify an Inverse Matrix

18. ANS: D

May want to write this as a system of equations, rather than in matrix form. But it still works.

PTS: 1 DIF: Advanced REF: Lesson 6-3

OBJ: 6-3.1 Solve systems of linear equations using inverse matrices.

NAT: 2 STA: 8.D.5 TOP: Solving Linear Systems Using Inverses and Cramer's Rule

KEY: Matrix Equations | Systems of Equations NOT: Example 1: Multiply Matrices

19. ANS: A

Feedback
A / Correct!
B / Substitute x = –5, y = –10, and z = –8 back into each equation in the system.
C / The system has a unique solution.
D / Substitute x = –5, y = 3, and z = 9 back into each equation in the system.

PTS: 1 DIF: Average REF: Lesson 6-3

OBJ: 6-3.1 Solve systems of linear equations using inverse matrices.

NAT: 2 STA: 8.D.5 TOP: Solving Linear Systems Using Inverses and Cramer's Rule

KEY: Matrices | Inverse Matrices | Systems of Linear Equations

NOT: Example 2: Solve a 3x3 System Using an Inverse Matrix

SHORT ANSWER

20. ANS:

PTS: 1

21. ANS:

PTS: 1

22. ANS:

PTS: 1

23. ANS:

3

PTS: 1

24. ANS:

-42

PTS: 1

25. ANS:

PTS: 1