Deck 10: Matrices and Determinants

Question 1

Fill in the blank using elementary row operations to form a row-equivalent matrix.

A)
B)
C)
D)
E)

Answer 1

Question 2

Perform the sequence of row operations on the matrix. What did the operations accomplish? Show all work.
(a) Add -4 times R₁ to R₂.
(b) Add -4 times R₁ to R₃.
(c) Add 2 times R₂ to R₃.
(d) Add 1 times R₂ to R₁.

Answer 2

(a) Add -4 times R₁ to R₂. (b) Add -4 times R₁ to R₃. (c) Add 2 times R₂ to R₃. (d) Add 1 times R₂ to R₁. The operations produce a row-equivalent matrix in reduced row-echelon form.

Question 3

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices.

A) x = -8, y = -4, z = 7
B) x = 8, y = 4, z = 7
C) x = -4, y = 7, z = 8
D) The systems yield different solutions.
E) x = 4, y = 7, z = 8

Answer 3

The systems yield different solutions.

Question 4

Determine the order of the matrix.

A) 1 × 1
B) 3
C) 3 × 3
D) 3 × 1
E) 1 × 3

Question 5

Write the system of linear equations represented by the augmented matrix. (Use variables x, y, z, and w.)

A)
B)
C)
D)
E)

Question 6

Find x and y.

A) x = 2, y = 5
B) x = 2, y = 2
C) x = -2, y = -5
D) x = 5, y = 2
E) x = -3, y = 1

Question 7

Use a system of equations to find the specified equation that passes through the points. Solve the system using matrices. Parabola: y = ax² + bx + c

A)
B)
C)
D)
E)

Question 8

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

A) x = 5, y = -1
B) x = 1, y = -5
C) x = -1, y = 5
D) x = 5, y = 1
E) no solution

Question 9

Identify the elementary row operation being performed to obtain the new row-equivalent matrix.

A) Add 2 times R₁ to R₂.
B) Add -2 times R₁ to R₂.
C) Add 2 times R₂ to R₁.
D) Add -2 times R₂ to R₁.
E) Add 2 times R₁ to R₁.

Question 10

Determine the order of the matrix.

A) 3 × 3
B) 2 × 2
C) 3 × 1
D) 2 × 3
E) 3 × 2

Question 11

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices.

A) x = -3, y = -6, z = 2
B) The systems yield different solutions.
C) x = 3, y = 2, z = -3
D) x = 3, y = 6, z = 2
E) x = -6, y = 2, z = -3

Question 12

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

A) x = -5, y = -3, z = 2
B) x = 3, y = -5, z = -2
C) x = 5, y = -3, z = 2
D) no solution
E) x = 5, y = 3, z = 2

Question 13

Determine whether the matrix is in row-echelon form. If it is, determine if it is also in reduced row-echelon form.

A) row-echelon form and reduced row-echelon form
B) row-echelon form
C) neither

Question 14

An augmented matrix that represents a system of linear equations (in variables x, y, and z) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix.

A) x = 0, y = 0, z = 0
B) x = -5x, y = -4y, z = -3z
C) x = -5, y = -3, z = -4
D) x = 5, y = 3, z = 4
E) x = -5, y = 0, z = 0

Question 15

Use the matrix capabilities of a graphing utility to write the matrix in reduced row-echelon form.

A)
B)
C)
D)
E)

Question 16

The currents in an electrical network are given by the solutions of the system where I₁, I₂, and I₃ are measured in amperes. Solve the system of equations using matrices.

A) no solution
B) I₁ = 3, I₂ = 5, I₃ = 9
C) I₁ = 5, I₂ = 3, I₃ = 8
D) I₁ = 3, I₂ = 5, I₃ = 8
E) I₁ = 5, I₂ = 8, I₃ = 3

Question 17

Write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables x, y, and z.)

A) x = 2, y = 3, z = 1
B) x = -1, y = -2, z = 3
C) x = -1, y = 2, z = -3
D) x = 83, y = -5, z = -4
E) x = -2, y = -1, z = 3

Question 18

Write the augmented matrix for the system of linear equations.

A)
B)
C)
D)
E)

Question 19

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. Show all of your work.

Question 20

Write the matrix in reduced row-echelon form.

A)
B)
C)
D)
E)

Question 21

If possible, find AB.

A)
B) not possible
C)
D)
E)

Question 22

Evaluate the expression.

A)
B)
C)
D) not possible
E)

Question 23

Write the system of linear equations as a matrix equation AX = B, and use Gauss-Jordan elimination on the augmented matrix to solve for the matrix X. Show all your work.

Question 24

Use the matrix capabilities of a graphing utility to find AB, if possible.

A)
B)
C)
D)
E) not possible

Question 25

Of the products AB, BA, A², and B², which ones are possible for the given matrices?

A) AB only
B) AB and BA only
C) BA only
D) A² and B² only
E) AB, BA, A², and B²

Question 26

Use the matrix capabilities of a graphing utility to find AB, if possible.

A)
B)
C)
D) not possible
E)

Question 27

Solve for X in the equation given.

A)
B)
C)
D)
E)

Question 28

Show that A is the inverse of
B. Show all your work. ALERT TO PUBLISHER: Diploma contained variable "b" and a condition involving "b" that is not used in this problem. I deleted these from the question.

Question 29

Evaluate the expression.

A)
B)
C) not possible
D)
E)

Question 30

Write the system of linear equations as a matrix equation AX = B, and use Gauss-Jordan elimination on the augmented matrix to solve for the matrix X. Show all your work.

Question 31

Evaluate the expression.

A)
B)
C)
D) not possible
E)

Question 32

Solve for X in the equation given.

A)
B)
C) not possible
D)
E)

Question 33

Show that B is the inverse of
A. Show all your work.

Question 34

Evaluate the expression.

A)
B)
C) not possible
D)
E)

Question 35

If possible, find 2A - 5B.

A)
B) not possible
C)
D)
E)

Question 36

If possible, find AB.

A)
B)
C)
D)
E) not possible

Question 37

Use the matrix capabilities of a graphing utility to evaluate the expression.

A)
B)
C)
D) not possible
E)

Question 38

Of the products AB, BA, A², and B², which ones are possible for the given matrices?

A) B² only
B) A² only
C) BA only
D) AB only
E) AB and BA only

Question 39

Find x and y.

A) x = -3, y = 3
B) x = 3, y = -4
C) no solution
D) x = 2, y = -3
E) x = 2, y = -4

Question 40

A lawn-and-garden store sells three types of fertilizer in 50-pound bags: Feed & Weed, Winterizer Blend, and 17-17-17. The number of bags that were sold last weekend is represented by A. The selling price per bag and the profit per bag for the three types of fertilizer are represented by B (values are in dollars).
(a) Compute AB and interpret the result. Show all your work.
(b) Find the store's total profit from fertilizer sales for the weekend. Show your work.

Question 41

Use the matrix capabilities of a graphing utility to find the determinant of the matrix .

A) -0.027
B) -0.18
C) -0.135
D) 0.081
E) -0.054

Question 42

Solve the system of linear equations using the inverse matrix .

A)
B)
C)
D)
E)

Question 43

Find the inverse of the matrix .

A)
B)
C)
D)
E)

Question 44

Use the matrix capabilities of a graphing utility to solve the following system of linear equations:

A)
B)
C)
D)
E)

Question 45

Use the matrix capabilities of a graphing utility to solve the following system of linear equations:

A)
B)
C)
D)
E)

Question 46

Find the inverse of the matrix (if it exists).

A) does not exist
B)
C)
D)
E)

Question 47

Solve the system of linear equations using the inverse matrix .

A)
B)
C)
D)
E)

Question 48

Find the determinant of the matrix .

A) 6
B) -12
C) -63
D) -33
E) 22

Question 49

Find the determinant of the matrix .

A)
B)
C)
D)
E)

Question 50

Use the matrix capabilities of a graphing utility to find the inverse of the matrix (if it exists).

A)
B)
C)
D) does not exist
E)

Question 51

Find all (a) minors and (b) cofactors of the matrix . Show your work.

Question 52

Solve the system of linear equations using the inverse matrix

A)
B)
C)
D)
E)

Question 53

Find the inverse of the matrix (if it exists).

A)
B) does not exist
C)
D)
E)

Question 54

Solve the system of linear equations using an inverse matrix.

A)
B)
C)
D)
E)

Question 55

Find the inverse of the matrix .

A)
B)
C)
D)
E)

Question 56

Find all (a) minors and (b) cofactors of the matrix . Show your work.

Question 57

Solve the system of linear equations using the inverse matrix .

A)
B)
C)
D)
E)

Question 58

Solve the system of linear equations using the inverse matrix .

A)
B)
C)
D)
E)

Question 59

Solve the system of linear equations using the inverse matrix .

A)
B)
C)
D)
E)

Question 60

Solve the system of linear equations using an inverse matrix.

A)
B)
C)
D)
E)

Question 61

Solve for x given the following equation involving a determinant.

A)
B)
C)
D)
E)

Question 62

Solve for x given the following equation involving a determinant.

A)
B)
C)
D)
E)

Question 63

Find the determinant of by the method of expansion by cofactors using column 3. Show your work.

Question 64

Use the matrix capabilities of a graphing utility to find the determinant of the matrix .

A) -12
B) -10
C) 7
D) -6
E) 4

Question 65

Given and , find .

A) -182,250
B) -153,090
C) 218,700
D) 273,375
E) 290,871

Question 66

Evaluate the determinant in which the entries are functions.

A)
B)
C)
D)
E)

Question 67

Use a determinant to find the area of the triangle shown below.

A) 14 square units
B) square units
C) square units
D) 12 square units
E) square units

Question 68

Use the matrix capabilities of a graphing utility to find the determinant of the matrix .

A) -432
B)
C)
D) -1080
E)

Question 69

Evaluate the determinant of the matrix below to determine which of the following makes the equation true.

A)
B)
C)
D)
E)

Question 70

Use Cramer's Rule to solve the following system of linear equations:

A)
B)
C)
D)
E)

Question 71

Find the determinant of by the method of expansion by cofactors using row 2. Show your work.

Question 72

You inherited a triangular piece of property after your Uncle Izzy passed away. You want to know the size of land, so you "step it off" to estimate the square footage. From the southernmost vertex A, you travel north 220 feet then west 280 feet (for vertex C), and from the southernmost vertex A, you travel 520 feet north then 110 feet west (for vertex B). Use a graphing utility to approximate the number of square feet of land that you have inherited.

A) 16,500 square feet
B) 60,700 square feet
C) 84,900 square feet
D) 59,400 square feet
E) 2200 square feet

Question 73

Evaluate the determinant in which the entries are functions.

A)
B)
C)
D)
E)

Question 74

Given , find .

A) 168
B) -80
C) 88
D) 40
E) -120

Question 75

A triangular region of farmland has become overrun with deer and will be open to the public for hunting to reduce the population. In order to know how many hunters to allow on the land at one time, you need to know the area of the region in square miles. In order to estimate the area of the region, you travel from the southernmost vertex C north 24 miles then west 20 miles (for vertex B), and from the southernmost vertex C you travel 56 miles north then 11 miles east (for vertex A). Use a graphing utility to approximate the number of square miles of land.

A) 428 square miles
B) 68 square miles
C) 176 square miles
D) 548 square miles
E) 692 square miles

Question 76

Determine a positive value for y such that a triangle with vertices , and has an area of 12 square units.

A) 11
B) 8
C) 6
D) 9
E) 1

Question 77

Use Cramer's Rule to solve the following system of linear equations:

A)
B)
C)
D)
E)

Question 78

Find the determinant of .

A) 64
B) -88
C) -110
D) -96
E) 0

Question 79

Use the matrix capabilities of a graphing utility to find the determinant of the matrix .

A) -48
B) 128
C) -64
D) -32
E) -384

Question 80

Find the determinant of .

A) -4
B)
C) -3
D) 7
E) 8