Deck 6: Matrices and Determinants

Question 1

Write the augmented matrix for the system of linear equations.

A)
B)
C)
D)
E)

Answer 1

B

Question 2

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

A) x = -2, y = 8, z = -3
B) x = 2, y = -8, z = -3
C) x = -8, y = -3, z = -2
D) x = 8, y = -3, z = -2
E) The systems yield different solutions.

Answer 2

A

Question 3

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

A) x = -6, y = 2, z = 5
B) x = 6, y = 2, z = 5
C) x = 6, y = -2, z = 5
D) x = -2, y = -6, z = -5
E) no solution

Answer 3

C

Question 4

Solve the system of equations below by the Gaussian elimination method:

A)
B)
C)
D)
E) no solution

Question 5

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

Question 6

Write the augmented matrix of the system of equations below.

A)
B)
C)
D)
E)

Question 7

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 = -8, y = 1, z = -1
B) x = -2, y = -4, z = -5
C) x = -4, y = 2, z = 5
D) x = -4, y = -2, z = -5
E) x = 2, y = -5, z = 4

Question 8

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

A) (19a + 105, -3a - 17, a) where a is any real number
B) (-9a - 11, -7a - 35, a) where a is any real number
C) (-12a + 96, -12a + 37, a) where a is any real number
D) (-125a + 105, 6a - 17, a) where a is any real number
E) (-17a - 33, 6a - 17, a) where a is any real number

Question 9

Identify the elementary row operation being performed to obtain the new row-equivalent matrix. Original Matrix New Row-Equivalent Matrix

A) Add 4 times R₁ to R₂.
B) Add -4 times R₂ to R₁.
C) Add -4 times R₁ to R₂.
D) Add 4 times R₁ to R₁.
E) Add 4 times R₂ to R₁.

Question 10

Determine whether the matrix is in row-echelon form and/or in reduced row-echelon form.

A) The matrix is in row-echelon form but not in reduced row-echelon form.
B) The matrix is in reduced row-echelon form.
C) The matrix is not in row-echelon or in reduced-row echelon form.
D) The matrix is not in row-echelon form but is in reduced row-echelon form.

Question 11

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

A)
B)
C)
D)
E)

Question 12

Write the matrix in reduced row-echelon form.

A)
B)
C)
D)
E)

Question 13

Solve the system of equations below by the Gaussian elimination method.

A)
B)
C)
D)
E) no solution

Question 14

Use elementary row operations to write the matrix below in row echelon form.

A)
B)
C)
D)
E)

Question 15

Solve the system of equations below by the Gaussian elimination method:

A)
B)
C)
D)
E) no solution

Question 16

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

A) x = -2, y = 5, z = 3
B) x = 2, y = -5, z = 3
C) x = -5, y = 3, z = -2
D) x = 5, y = 3, z = -2
E) The systems yield different solutions.

Question 17

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 = 8, y = 1, z = 5
B) x = -8, y = -1, z = -5
C) x = 0, y = 0, z = 0
D) x = 8x, y = 5y, z = z
E) x = 8, y = 0, z = 0

Question 18

Determine the order of the matrix below.

A) 6
B)
C)
D) 5
E)

Question 19

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

A)
B)
C)
D)
E)

Question 20

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

A) x = -2, y = 1
B) x = 2, y = -1
C) x = -1, y = 2
D) x = -1, y = -2
E) no solution

Question 21

Find x and y.

A) x = -3, y = 5
B) x = 8, y = -1
C) x = -5, y = 3
D) x = 3, y = -5
E) x = 3, y = 3

Question 22

Evaluate the expression.

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

Question 23

Find

A)
B)
C)
D)
E)

Question 24

If possible, find AB.

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

Question 25

Find x and y.

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

Question 26

Solve for X in the equation given.

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

Question 27

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

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

Question 28

Solve for X in the equation given.

A)
B)
C)
D)
E)

Question 29

If possible, find AB.

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

Question 30

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

A)
B)
C)
D)
E)

Question 31

Find

A)
B)
C)
D)
E)

Question 32

Evaluate the expression.

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

Question 33

Find

A)
B)
C)
D)
E)

Question 34

Find

A)
B)
C)
D)
E)

Question 35

If possible, find 4A + 2B.

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

Question 36

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

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

Question 37

If possible, find A - B.

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

Question 38

A city zoo borrowed $2,000,000 at simple annual interest to construct a breeding facility.Some of the money was borrowed at 9%, some at 11%, and some at 14%.Use a system of equations to determine how much was borrowed at 9% if the total annual interest was $216,000 and the amount borrowed at 9% was twice the amount borrowed at 14%.Solve the system using matrices.

A) $400,000 was borrowed at 9%
B) $800,000 was borrowed at 9%
C) $300,000 was borrowed at 9%
D) $1,600,000 was borrowed at 9%
E) $1,700,000 was borrowed at 9%

Question 39

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

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

Question 40

Find A².(Note: A² = AA.)

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

Question 41

Find

A)
B)
C)
D)
E)

Question 42

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

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

Question 43

A convention planning service has identified three suitable hotels for a convention.The quoted room rates are for single, double, triple, and quadruple occupancy.The current cost for each type of room at each hotel is represented by the matrix where the columns correspond to each hotel and the rows correspond to single, double, triple, and quadruple occupancy respectively.If room rates are guaranteed not to increase by more than 5% by the time of the convention, what is the maximum rate per room per hotel?

A)
B)
C)
D)
E)

Question 44

Find the inverse of the matrix

A)
B)
C)
D)
E)

Question 45

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

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

Question 46

Given matrix Find the inverse matrix.

A)
B)
C)
D)
E)

Question 47

Find the inverse of the matrix below, if it exists.

A)
B)
C)
D)
E) The inverse of the matrix does not exist.

Question 48

Solve the system of linear equations using an inverse matrix.

A)
B)
C)
D)
E)

Question 49

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

A)
B)
C)
D)
E)

Question 50

Write the system of linear equations as a matrix equation and then use Gauss-Jordan elimination on the augmented matrix to solve for the the matrix

A)
B)
C)
D)
E)

Question 51

Find which of the following matrices is an inverse to the matrix A below if any. Matrix I : Matrix II : Matrix III :

A) Each of the matrices I and III is an inverse of A.
B) Each of the matrices I and II is an inverse of A.
C) Each of the matrices II and III is an inverse of A.
D) Matrix II is an inverse of A.
E) None of the matrices is an inverse of A.

Question 52

Find the inverse of the matrix

A)
B)
C)
D)
E)

Question 53

You invest in AAA-rated bonds, A-rated bonds, and B-rated bonds.Your average yield is 6% on AAA bonds, 9% on A bonds, and 9% on B bonds.You invest twice as much in B bonds as in A bonds.The desired system of linear equations (where and represent the amounts invested in AAA, A, and B bonds, respectively) is as follows. Use the inverse of the coefficient matrix of this system to find the amount invested in A bonds for the given a total investment of $45,000 and annual return of $3780.

A) $12,000
B) $24,000
C) $10,000
D) $1000
E) $6000

Question 54

Find the inverse of the given matrix.

A)
B)
C)
D)
E) The matrix does not have an inverse.

Question 55

A fruit grower raises apples and peaches, which are shipped to three different outlets.The number of units of each fruit that is shipped to each outlet is shown in the matrix where the first row contains the data for the number of apples shipped, the second row contains the data for the number of peaches shipped, and the columns correspond to outlets X, Y, and Z respectively.The profit per unit of apples is $4.50 and the profit per unit of peaches is $5.00.Organize the profits per unit in a matrix Compute to find the profits from both crops at outlet

A) $1075.00
B) $587.50
C) $500.00
D) $1050.00
E) $1575.00

Question 56

Find A².(Note: A² = AA.)

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

Question 57

Two competing companies offer cable television to a city with 100,000 households.Gold Cable Company has 25,000 subscribers and Galaxy Cable Company has 15,000 subscribers.The percent changes in cable subscriptions each year are shown in the matrix below where the columns are percent changes from Gold, from Galaxy, and from Nonsubscriber respectively and the rows are percent changes to Gold, to Galaxy, and to Nonsubscriber respectively.Find the number of subscribers Gold Cable will have in two years using matrix multiplication.

A) 45,250
B) 22,250
C) 32,500
D) 36,040
E) 36,250

Question 58

Solve the system of linear equations using an inverse matrix.

A)
B)
C)
D)
E)

Question 59

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

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

Question 60

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

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

Question 61

Find the determinant of

A)
B)
C)
D)
E)

Question 62

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

Question 63

Given , find

A)
B)
C)
D)
E)

Question 64

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

A)
B)
C)
D)
E)

Question 65

Find the determinant of the matrix

A)
B)
C)
D)
E)

Question 66

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

Question 67

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

A)
B)
C)
D)
E)

Question 68

Find the determinant of

A)
B)
C)
D)
E)

Question 69

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

Question 70

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

A)
B)
C)
D)
E)

Question 71

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

A)
B)
C)
D)
E)

Question 72

Find the determinant of by the method of expansion by cofactors.

A)
B)
C)
D)
E)

Question 73

Find the determinant of the matrix

A)
B)
C)
D)
E)

Question 74

Consider the circuit shown in the figure below.The currents ( in amperes ) are the solutions to the system of linear equations , where are voltages of 9 and 9 volts, respectively.Use the inverse of the coefficient matrix of this system to find the unknown currents.Round answers to nearest tenth.[Hint: If a current value turns out to be negative, it simply means that the current flow is in the opposite direction from that indicated in the figure below.]

A)
B)
C)
D)
E)

Question 75

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

A)
B)
C)
D)
E)

Question 76

Which of the following upper triangular matrices has a determinant equal to 16.

A) matrix D
B) matrix B
C) matrix A
D) matrix C
E) matrix E

Question 77

Evaluate the determinant of the matrix below.

A) 3
B) -2
C) 5
D) 24
E) 32

Question 78

Find the determinant of the matrix below.Expand by cofactors on the row or column that appears to make the computations easiest.

A) -20
B) -216
C) 120
D) 456
E) 180

Question 79

Evaluate the determinant by expanding by cofactors.

A)
B)
C)
D)
E)

Question 80

Given and , find

A)
B)
C)
D)
E)