Deck 2: Systems of Linear Equations and Matrices

Question 1

An m × n zero matrix serves as an m × n .

Answer 1

additive identity

Question 2

Answer 2

Question 3

Use a graphing calculator to solve the system of equations. Round your solution to one decimal place. $$2.7 x-0.2 y-5.0 z=2.6$$
$5.6 x+4.6 y-0.5 z=-4.1$
$3.4 x-1.3 y+1.6 z=10.5$

A) $(1.7,-2.9,0.5)$
B) $(8.6,-14.6,2.6)$
C) $(6.9,-11.7,2.1)$
D) $(3.4,-5.9,1.1)$

Answer 3

$(1.7,-2.9,0.5)$

Question 4

Provide an appropriate response
Which choice best describes the following matrix?

Question 5

Perform the indicated operation, where possible.

Question 6

Use the Gauss-Jordan method to solve the system of equations. $$\begin{array}{l}3 x-2 y=-3 \\9 x-6 y=-9\end{array}$$

A) $(1,3)$
B) $\left(-1-\frac{2}{3} y, y\right)$
C) $\left(-1+\frac{2}{3} y, y\right)$
D) No solution

Question 7

Solve the problem.
16)

Question 8

Does a matrix with a column of all zeros have an inverse? Why or why not?

Question 9

Use the Gauss-Jordan method to solve the system of equations.

Question 10

The property does not apply to matrix multiplication.

Question 11

Question 12

If a variable is expressed in terms of another variable, what is the other variable called?

Question 13

How many solutions are there to a dependent system?

Question 14

Describe the proper form for the system to be in before the Gauss-Jordan method can be used.

Question 15

Use a graphing calculator to solve the system of equations. Round your solution to one decimal place. $$\begin{array}{l}1.5 x-0.4 y+1.6 z=2.4 \\4.5 x-7.0 y-0.4 z=-4.0 \\3.8 x+2.4 y+2.0 z=8.6\end{array}$$

A) $(0.8,0.9,0.6)$
B) $(1.1,1.2,0.8)$
C) $(0.5,0.6,0.4)$
D) $(0.3,0.3,0.2)$

Question 16

Use the echelon method to solve the system. $$\begin{array}{l}\frac{x}{5}+\frac{y}{5}=1 \\\frac{x}{5}-\frac{y}{5}=-\frac{7}{5}\end{array}$$

A) $(1,7)$
B) $(-2,7)$
C) $(-1,6)$
D) No solution

Question 17

Can non-square matrices have inverses? Why or why not?

Question 18

Use the Gauss-Jordan method to solve the system of equations. $$\begin{array}{l}-2 x-8 y=6 \\-4 x-16 y=2\end{array}$$

A) $(2,2)$
B) $\left(-\frac{4}{3} x-\frac{1}{3} y, y\right)$
C) $(6,2)$
D) No solution

Question 19

Use the Gauss-Jordan method to solve the system of equations. $8 w+8 x-6 y-2 z=-30$
$7 w+6 x-9 y-2 z=-38$
$8 w+8 x+7 y+3 z=17$
$-6 w-2 x+8 y+8 z=18$

A) (-1,4,-5,2)
B) (2,3,-4,-2)
C) (2,-3,4,-1)
D) No solution

Question 20

Is this a square matrix? Why or why not?

Question 21

Find the matrix product, if possible.

Question 22

Solve the problem.
A company makes three chocolate candies: cherry, almond, and raisin. Matrix A gives the amount of ingredients in one batch. Matrix B gives the costs of ingredients from suppliers J and K. What is
The cost of 100 batches of each candy using ingredients from supplier J?

Question 23

Solve the problem.

-A basketball fieldhouse seats 15,000. Courtside seats cost $10, endzone seats cost $6, and balcony seats cost $4. The total revenue for a sellout is $82,000. If half the courtside seats, half the balcony Seats, and all the endzone seats are sold; then the total revenue is $47,000. How many of each type Of seat are there?

A)4000 courtside, 3000 endzone, 8000 balcony
B)3000 courtside, 4000 endzone, 8000 balcony
C)3000 courtside, 2000 endzone, 10,000 balcony
D)3200 courtside, 1800 endzone, 10,000 balcony

Question 24

Decide whether the matrices are inverses of each other. (Check to see if their product is the identity matrix I.) $\left.\begin{array}{rr}-2 & 4 \\ 4 & -4\end{array}\right]$ and $\left[\begin{array}{ll}\frac{1}{2} & \frac{1}{4} \\ \frac{1}{2} & \frac{1}{4}\end{array}\right]$

Question 25

Find the matrix product, if possible.

Question 26

Use the Gauss-Jordan method to solve the system of equations.

Question 27

Solve the system of equations. Let z be the parameter.

Question 28

Perform the indicated operation, where possible.

Question 29

Solve the problem.

-A company makes three chocolate candies: cherry, almond, and raisin. Matrix A gives the amount of ingredients in one batch. Matrix B gives the costs of ingredients from suppliers X and Y. What is The cost of 100 batches of each candy using ingredients from supplier X?
$$A=\left[\begin{array}{ccc}\text { sugar choc milk } \\4 & 6 & 1 \\5 & 3 & 1 \\3 & 3 & 1\end{array}\right] \text { cherry }$$
$$B=\left[\begin{array}{ll}X & Y \\3 & 2 \\3 & 4 \\2 & 2\end{array}\right] \text { sugar }$$

A) $\$ 4800$
B) $\$ 7800$
C) $\$ 3300$
D) $\$ 6600$

Question 30

Use the echelon method to solve the system of two equations in two unknowns.

Question 31

Decide whether the matrices are inverses of each other. (Check to see if their product is the identity matrix I.)

Question 32

Use the indicated row operation to change the matrix.

Question 33

Find the value.

Question 34

Write the system of equations associated with the augmented matrix.

Question 35

Use the echelon method to solve the system.

Question 36

Find the values of the variables in the equation.

Question 37

Solve the system of equations by using the inverse of the coefficient matrix if it exists and by the echelon method if the
inverse doesn't exist.

Question 38

Decide whether the matrices are inverses of each other. (Check to see if their product is the identity matrix I.) $\left[\begin{array}{rr}-5 & -1 \\ 6 & 0\end{array}\right]$ and $\left[\begin{array}{cc}0 & \frac{1}{6} \\ -1 & \frac{5}{6}\end{array}\right]$

Question 39

Use a graphing calculator to solve the system of equations. Round your solution to one decimal place. $\left[\begin{array}{rr}9 & -2 \\ 7 & -2\end{array}\right]$ and $\left[\begin{array}{rr}\frac{1}{2} & \frac{1}{2} \\ -\frac{7}{4} & -\frac{9}{4}\end{array}\right]$

Question 40

Solve the problem.
The matrices give points and rebounds for five starting players in two games. Find the matrix that gives the totals.

Question 41

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these
products exist.

Question 42

Use a graphing calculator to find the matrix product and/or sum.

Question 43

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these
products exist.

Question 44

Solve the system of equations by using the inverse of the coefficient matrix if it exists and by the echelon method if the
inverse doesn't exist.

Question 45

Solve the problem.

-An investor has $400,000 to invest in stocks, bonds, and commodities. If he plans to put three times as much into stocks as in bonds, how can he distribute his money among the three types of Investments? (Let x denote the amount put into stocks, y the amount put into bonds, and z the Amount put into commodities. Let all amounts be in dollars, and let z be the parameter.)

A) $x=300,000-z / 2, y=100,000-z / 2,0 \leq z \leq 400,000$
B) $x=100,000-z / 2, y=300,000-z / 2,0 \leq z \leq 400,000$
C) $x=300,000-3 z / 4, y=100,000-z / 4,0 \leq z \leq 400,000$
D) $x=100,000-3 z / 4, y=300,000-z / 4,0 \leq z \leq 400,000$

Question 46

Solve the problem.
Carney and Dobler sell auto and hazard insurance. Their sales, in dollars, for the months of July and August are given in the following matrices.

Question 47

The sizes of two matrices A and B are given. Find the sizes of the product AB and the product BA, whenever these products exist.

- $A$ is $3 \times 2$ , and B is $2 \times 3$ .

A) $2 \times 2 ; 3 \times 3$
B) $3 \times 3$ ; BA does not exist.
C) $\mathrm{AB}$ does not exist: $2 \times 2$
D) $3 \times 3 ; 2 \times 2$

Question 48

Find the inverse, if it exists, for the matrix.

Question 49

Find the matrix product, if possible.

Question 50

Solve the problem.
Anne and Nancy use a metal alloy that is 25.75% copper to make jewelry. How many ounces of a 19% alloy must be mixed with a 28% alloy to form 92 ounces of the desired alloy?

A)25 ounces
B)23 ounces
C)74 ounces
D)69 ounces

Question 51

Find the inverse, if it exists, for the matrix.

Question 52

Write a matrix to display the information.

-A bakery sells three types of cakes. Cake I requires 2 cups of flour, 2 cups of sugar, and 2 eggs. Cake II requires 4 cups of flour, 1 cup of sugar, and 1 egg. Cake III requires 2 cups of flour, 2 cups of Sugar, and 3 eggs. Make a 3 × 3 matrix showing the required ingredients for each cake. Assign the
Cakes to the rows and the ingredients to the columns.

A) $\left[\begin{array}{lll}2 & 1 & 3 \\ 2 & 1 & 2 \\ 2 & 4 & 2\end{array}\right]$
B) $\left[\begin{array}{lll}2 & 4 & 2 \\ 2 & 1 & 2 \\ 2 & 1 & 3\end{array}\right]$
C) $\left[\begin{array}{lll}2 & 2 & 2 \\ 4 & 1 & 1 \\ 2 & 2 & 3\end{array}\right]$
D) $\left[\begin{array}{lll}2 & 2 & 3 \\ 1 & 1 & 4 \\ 2 & 2 & 2\end{array}\right]$

Question 53

Find the ratios of products A, B, and C using a closed model.

Question 54

Find the matrix product, if possible.

Question 55

Provide an appropriate response.
What are the elements of the third row of the following matrix? 42)

Question 56

Decide whether the matrices are inverses of each other. (Check to see if their product is the identity matrix I.)

- $\left[\begin{array}{ll}-5 & 1 \\ -7 & 1\end{array}\right]$ and $\left[\begin{array}{cc}\frac{1}{2} & -\frac{1}{2} \\ \frac{7}{2} & -\frac{5}{2}\end{array}\right]$

Question 57

Solve the system of equations by using the inverse of the coefficient matrix.

Question 58

Find the ratios of products A, B, and C using a closed model.

Question 59

Find the matrix product, if possible.

Question 60

Find the inverse, if it exists, for the matrix.

Question 61

Use a graphing calculator to solve the system of equations. Round your solution to one decimal place.

Question 62

Use the Gauss-Jordan method to solve the system of equations.

Question 63

Find the ratios of products A, B, and C using a closed model.

-

A) 13: 25: 100
B) $33: 67: 63$
C) 1: $149: 100$
D) 1: 25: 25

Question 64

Solve the system of equations by using the inverse of the coefficient matrix if it exists and by the echelon method if the inverse doesn't exist.

- $$\begin{array}{l}3 x+y=15 \\2 x+4 y=0\end{array}$$

A) $(-3,-6)$
B) $(6,-3)$
C) $(-3,6)$
D) No inverse, no solution for system

Question 65

Find the production matrix for the input-output and demand matrices using the open model.

Question 66

Solve the problem.

-A company makes three chocolate candies: cherry, almond, and raisin. Matrix A gives the amount of ingredients in one batch. Matrix B gives the costs of ingredients from suppliers J and K. What is The cost of 100 batches of each candy using ingredients from supplier K?

A) $\$ 12.000$
B) $\$ 15.200$
C) $\$ 6000$
D) $\$ 9400$

Question 67

Solve the problem.
Suppose the following matrix represents the input-output matrix, T, of a simplified economy. Find the amount of each commodity that should be produced.

A)3360 units of manufacturing, 2796 units of agriculture, and 3012 units of transportation.
B)2976 units of manufacturing, 3386.4 units of agriculture, and 2791.2 units of transportation.
C)3012 units of manufacturing, 3360 units of agriculture, and 2791.2 units of transportation.
D)3012 units of manufacturing, 2796 units of agriculture, and 2976 units of transportation.

Question 68

Question 69

Solve the system of equations by using the inverse of the coefficient matrix.

Question 70

Find the ratios of products A, B, and C using a closed model.

-

A) $4: 8: 3$
B) $3: 2: 3$
C) $8: 16: 3$
D) $3: 3: 2$

Question 71

AB is a 2 × 2 matrix.

Question 72

Use a graphing calculator to find the matrix product and/or sum.

Question 73

Use the Gauss-Jordan method to solve the system of equations.

Question 74

For the following systems of equations in echelon form, tell how many solutions there are in nonnegative integers.

Question 75

Find the inverse, if it exists, for the matrix.

Question 76

Find the ratios of products A, B, and C using a closed model.

Question 77

Solve the system of equations by using the inverse of the coefficient matrix.

Question 78

Provide an appropriate response.
True or False?

Question 79

Use the indicated row operation to change the matrix.

Question 80

Use a graphing calculator to find the matrix product and/or sum.
Find BA.