Matrices and Matrix Operations Practice Questions — ACT

Question 1Question →

Let matrix $A = \begin{pmatrix} 2 & 5 \\ -1 & 3 \end{pmatrix}$ and matrix $B = \begin{pmatrix} 4 & -2 \\ 3 & 1 \end{pmatrix}$ . If matrix $C = 2A + 3B$ , what is the value of the entry in the second row and first column of matrix $C$ ?

Show answer & explanation

Answer: 7

Answer

The entry in the second row and first column of matrix

C

is 7.

The entry in the second row, first column of matrix

C

, denoted as

c_{2,1}

, is calculated by multiplying the corresponding entry in matrix

A

(

a_{2,1} = -1

) by 2 and the corresponding entry in matrix

B

(

b_{2,1} = 3

) by 3, and then adding the results:

2(-1) + 3(3) = -2 + 9 = 7

.

Step-by-Step Solution

1

Locate the element in the second row, first column of both matrices

A

and

B

.

The element in the second row, first column of

A

(designated

a_{2,1}

) is

-1

, and the element in the second row, first column of

B

(designated

b_{2,1}

) is

3

.

To find a specific entry in the resulting matrix

C = 2A + 3B

, we perform the scalar multiplication and addition operations on the corresponding entries in the same position of matrices

A

and

B

.

2

Multiply the corresponding elements by their respective scalar factors.

For matrix

A

,

2 \times (-1) = -2

. For matrix

B

,

3 \times 3 = 9

.

Scalar multiplication requires multiplying each individual entry of a matrix by the given scalar factor.

3

Add the two resulting values to find the entry for matrix

C

.

-2 + 9 = 7

Matrix addition is performed by adding the corresponding elements of the matrices.

Key Concept

Matrix scalar multiplication and addition of corresponding elements

Estimated Time:45s

Question 2Question →

A local bakery sells chocolate chip cookies for $1.50 each and oatmeal raisin cookies for$ 1.25 each. The table below shows the number of cookies sold on Monday and Tuesday:

Day	Chocolate Chip	Oatmeal Raisin
Monday	$80$	$60$
Tuesday	$95$	$70$

If the sales data is represented by the matrix $S = \begin{bmatrix} 80 & 60 \\ 95 & 70 \end{bmatrix}$ and the prices are represented by the matrix $P = \begin{bmatrix} 1.50 \\ 1.25 \end{bmatrix}$ , which of the following matrices represents the total revenue, in dollars, generated on Monday and Tuesday, respectively?

\begin{bmatrix} 190.00 \\ 223.75 \end{bmatrix}

\begin{bmatrix} 120.00 & 75.00 \\ 142.50 & 87.50 \end{bmatrix}

\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}

\begin{bmatrix} 238.75 \\ 177.50 \end{bmatrix}

\begin{bmatrix} 185.00 \\ 220.00 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}$

Answer

The matrix representing the revenue is

\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}

The correct matrix is obtained by performing standard matrix multiplication of

S

(dimensions

2 \times 2

) and

P

(dimensions

2 \times 1

). The resulting matrix has dimensions

2 \times 1

. The first element is the sum of the products of the first row of

S

and the column of

P

:

80 \times 1.50 + 60 \times 1.25 = 120 + 75 = 195

. The second element is the sum of the products of the second row of

S

and the column of

P

:

95 \times 1.50 + 70 \times 1.25 = 142.50 + 87.50 = 230

. This gives the matrix containing the values 195.00 and 230.00.

Step-by-Step Solution

1

Set up the matrix multiplication of sales and prices

Multiply the

2 \times 2

sales matrix

S

by the

2 \times 1

price matrix

P

:

S \times P = \begin{bmatrix} 80 & 60 \\ 95 & 70 \end{bmatrix} \begin{bmatrix} 1.50 \\ 1.25 \end{bmatrix}

Matrix multiplication aligns the quantities sold of each item with their respective unit prices to calculate the total revenue per day.

2

Perform the matrix multiplication for each row

Monday's revenue is

80 \times 1.50 + 60 \times 1.25 = 120 + 75 = 195

. Tuesday's revenue is

95 \times 1.50 + 70 \times 1.25 = 142.50 + 87.50 = 230

.

Multiplying the elements of each row in the first matrix by the corresponding elements in the column of the second matrix, and then summing them, calculates the dot product for each day.

3

Write the results in a

2 \times 1

matrix

The final product matrix is

\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}

.

The product of a

2 \times 2

matrix and a

2 \times 1

matrix is a

2 \times 1

matrix containing the total values for each row.

Key Concept

Matrix Multiplication in Word Problems

Question 3Question →

A local bakery sells two types of muffins: blueberry and bran. Store 1 sells 30 blueberry muffins and 20 bran muffins. Store 2 sells 40 blueberry muffins and 35 bran muffins. Blueberry muffins cost $3 each, and bran muffins cost$ 2 each. This information is represented by the inventory matrix $M$ and the price matrix $P$ :

M = \begin{pmatrix} 30 & 20 \\ 40 & 35 \end{pmatrix}

P = \begin{pmatrix} 3 \\ 2 \end{pmatrix}

Which of the following matrices represents the total revenue from muffin sales for Store 1 and Store 2, calculated using the matrix product $MP$ ?

\begin{pmatrix} 90 & 40 \\ 120 & 70 \end{pmatrix}

\begin{pmatrix} 120 \\ 185 \end{pmatrix}

\begin{pmatrix} 130 \\ 190 \end{pmatrix}

\begin{pmatrix} 130 & 190 \end{pmatrix}

\begin{pmatrix} 150 \\ 210 \end{pmatrix}

Show answer & explanation

Answer: $\begin{pmatrix} 130 \\ 190 \end{pmatrix}$

Answer

The column matrix with elements 130 and 190

The correct answer is the column matrix with elements 130 and 190. Multiplying the first row of the inventory matrix by the price column matrix yields the revenue for Store 1 (

30 \times 3 + 20 \times 2 = 130

). Multiplying the second row by the price column matrix yields the revenue for Store 2 (

40 \times 3 + 35 \times 2 = 190

). Since the product of a 2 by 2 matrix and a 2 by 1 matrix is a 2 by 1 column matrix, the final result is correctly formatted.

Step-by-Step Solution

1

Set up the matrix multiplication of the 2 by 2 matrix and the 2 by 1 matrix.

MP = \begin{pmatrix} 30 & 20 \\ 40 & 35 \end{pmatrix} \begin{pmatrix} 3 \\ 2 \end{pmatrix}

To calculate total revenue, we multiply the quantity matrix by the price matrix.

2

Calculate the first element of the resulting matrix (Store 1 total revenue) by taking the dot product of the first row of the inventory matrix and the price matrix.

30 \times 3 + 20 \times 2 = 90 + 40 = 130

This pairs the quantity of each muffin type sold at Store 1 with its respective price.

3

Calculate the second element of the resulting matrix (Store 2 total revenue) by taking the dot product of the second row of the inventory matrix and the price matrix.

40 \times 3 + 35 \times 2 = 120 + 70 = 190

This pairs the quantity of each muffin type sold at Store 2 with its respective price.

4

Assemble the resulting products into a 2 by 1 column matrix.

\begin{pmatrix} 130 \\ 190 \end{pmatrix}

The product of a 2 by 2 matrix and a 2 by 1 matrix is a 2 by 1 matrix.

Key Concept

Matrix multiplication of a 2 by 2 matrix and a 2 by 1 matrix in a real-world context.

Question 4Question →

A community theater sells adult tickets for $\$15$ each and student tickets for $\$10$ each. The number of tickets sold for the evening performances on Friday and Saturday is represented by the matrix $A$ :

A = \begin{bmatrix} 80 & 45 \\ 110 & 60 \end{bmatrix}

where the rows represent Friday and Saturday, respectively, and the columns represent adult and student tickets, respectively.

The ticket prices are represented by the matrix $B$ :

B = \begin{bmatrix} 15 \\ 10 \end{bmatrix}

The product matrix $C = AB$ represents the total ticket revenue for each night. What is the element in the second row and first column of matrix $C$ , which represents the total ticket revenue, in dollars, for Saturday's performance?

Show answer & explanation

Answer: 2250

Answer

The correct answer is

2250

dollars, which represents the ticket revenue for Saturday's performance.

The correct answer is

2250

dollars. The element in the second row and first column of the product matrix

C = AB

is obtained by multiplying the elements of the second row of matrix

A

(Saturday's ticket sales:

110

adult,

60

student) by the corresponding elements of the first column of matrix

B

(ticket prices:

\$15

adult,

\$10

student) and summing the products:

110 \times 15 + 60 \times 10 = 1650 + 600 = 2250

.

Step-by-Step Solution

1

Identify the row of matrix

A

representing Saturday's sales and the column of matrix

B

representing the prices.

The second row of matrix

A

is

\begin{bmatrix} 110 & 60 \end{bmatrix}

(representing

110

adult tickets and

60

student tickets sold on Saturday). Matrix

B

is

\begin{bmatrix} 15 \\ 10 \end{bmatrix}

(representing the prices of

\$15

and

\$10

respectively).

Since the question asks for Saturday's performance (the second row of matrix

A

) and the revenue is calculated by multiplying sales by prices, we use these specific components for the matrix multiplication.

2

Calculate the dot product of Saturday's sales row and the price column.

110 \times 15 + 60 \times 10 = 1650 + 600 = 2250

.

This calculation determines the sum of the revenue from adult tickets (

110 \times 15 = 1650

) and student tickets (

60 \times 10 = 600

) to yield the total revenue.

Key Concept

Matrix Multiplication

Question 5Question →

The matrices $A$ and $B$ are defined as follows:

A = \begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 1 \\ 0 & 2 \end{bmatrix}

What is the product matrix $AB$ ?

\begin{bmatrix} 6 & 2 \\ 3 & 7 \end{bmatrix}

\begin{bmatrix} 6 & 0 \\ 0 & 6 \end{bmatrix}

\begin{bmatrix} 7 & 3 \\ 2 & 6 \end{bmatrix}

\begin{bmatrix} 5 & 1 \\ 1 & 5 \end{bmatrix}

\begin{bmatrix} 6 & 0 \\ 6 & 6 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 6 & 2 \\ 3 & 7 \end{bmatrix}$

Answer

\begin{bmatrix} 6 & 2 \\ 3 & 7 \end{bmatrix}

The correct matrix is obtained by performing row-by-column matrix multiplication, which yields row 1 entries of 6 and 2, and row 2 entries of 3 and 7.

Step-by-Step Solution

1

Multiply the first row of

A

by the first column of

B

to find the element in the first row, first column of the product matrix.

2(3) + 0(0) = 6

The row-by-column rule dictates that the entry at row

i

, column

j

of the product is the dot product of row

i

of the first matrix and column

j

of the second matrix.

2

Multiply the first row of

A

by the second column of

B

to find the element in the first row, second column.

2(1) + 0(2) = 2

This determines the top-right entry of the resulting matrix.

3

Multiply the second row of

A

by the first column of

B

to find the element in the second row, first column.

1(3) + 3(0) = 3

This determines the bottom-left entry of the resulting matrix.

4

Multiply the second row of

A

by the second column of

B

to find the element in the second row, second column.

1(1) + 3(2) = 1 + 6 = 7

This determines the bottom-right entry of the resulting matrix.

Key Concept

Matrix multiplication involves computing the dot product of the rows of the first matrix with the columns of the second matrix.

Question 6Question →

A retail electronics store sells two models of wireless headphones: Solo, which sells for $80, and Studio, which sells for$ 150. During a holiday sale, the number of headphones sold at the Downtown and Uptown store locations is represented by the matrix $N$ below, where the first row represents the Downtown store, the second row represents the Uptown store, the first column represents Solo headphones, and the second column represents Studio headphones:

N = \begin{bmatrix} 40 & 25 \\ 30 & 35 \end{bmatrix}

The prices of the headphones are represented by the matrix $P$ below:

P = \begin{bmatrix} 80 \\ 150 \end{bmatrix}

Which of the following matrices represents the total revenue, in dollars, generated from headphone sales at each of the two stores, where the first row represents the Downtown store and the second row represents the Uptown store?

\begin{bmatrix} 8,000 \\ 7,300 \end{bmatrix}

\begin{bmatrix} 5,600 \\ 9,000 \end{bmatrix}

\begin{bmatrix} 6,950 \\ 7,650 \end{bmatrix}

\begin{bmatrix} 6,850 \\ 7,650 \end{bmatrix}

\begin{bmatrix} 3,200 & 3,750 \\ 2,400 & 5,250 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 6,950 \\ 7,650 \end{bmatrix}$

Answer

The column matrix with first row 6,950 and second row 7,650

The total revenue for each store is found by calculating the product of the sales matrix

N

and the price matrix

P

. This multiplication is performed by taking the dot product of each row in

N

with the column in

P

. For the first row (Downtown store), the calculation is

40 \times 80 + 25 \times 150 = 3,200 + 3,750 = 6,950

. For the second row (Uptown store), the calculation is

30 \times 80 + 35 \times 150 = 2,400 + 5,250 = 7,650

. Combining these results yields the column matrix containing 6,950 and 7,650.

Step-by-Step Solution

1

Set up the matrix multiplication to calculate the revenue, multiplying the sales matrix

N

by the price matrix

P

.

NP = \begin{bmatrix} 40 & 25 \\ 30 & 35 \end{bmatrix} \begin{bmatrix} 80 \\ 150 \end{bmatrix}

To find the total revenue for each store, the number of units sold (organized by store in rows and product type in columns) must be multiplied by the unit prices (organized by product type in rows).

2

Calculate the revenue for the Downtown store by taking the dot product of the first row of

N

and the column of

P

.

40 \times 80 + 25 \times 150 = 3,200 + 3,750 = 6,950

This combines the units sold of each headphone model at the Downtown store with their respective prices to find the store's total revenue.

3

Calculate the revenue for the Uptown store by taking the dot product of the second row of

N

and the column of

P

.

30 \times 80 + 35 \times 150 = 2,400 + 5,250 = 7,650

This combines the units sold of each headphone model at the Uptown store with their respective prices to find the store's total revenue.

4

Combine the results into the final product matrix.

\begin{bmatrix} 6,950 \\ 7,650 \end{bmatrix}

The resulting matrix contains the total revenues for the Downtown store in the first row and the Uptown store in the second row.

Key Concept

Matrix Multiplication and Modeling

Alternative Method

Instead of setting up the entire matrix multiplication, calculate the total revenue for the first store (Downtown) individually:

40 \times 80 + 25 \times 150 = 6,950

. This single calculation is enough to eliminate the options that do not have 6,950 in the first row.

Estimated Time:2m 0s

Question 7Question →

Matrices $A$ , $B$ , and $C$ are defined such that $A = \begin{bmatrix} x & 3 \\ -2 & 4 \end{bmatrix}$ , $B = \begin{bmatrix} 2 & -1 \\ y & 5 \end{bmatrix}$ , and $C = \begin{bmatrix} 14 & 11 \\ 4 & 22 \end{bmatrix}$ . If $AB = C$ , what is the value of $x + y$ ?

Show answer & explanation

Answer: 6

Answer

The value of

x + y

is 6.

The correct answer is 6 because performing the matrix multiplication

AB

yields the matrix

\begin{bmatrix} 2x + 3y & -x + 15 \\ -4 + 4y & 22 \end{bmatrix}

. Equating this to matrix

C

gives the equations

-x + 15 = 11

, which yields

x = 4

, and

-4 + 4y = 4

, which yields

y = 2

. The sum of these values is

4 + 2 = 6

. This is verified by checking the remaining equation

2(4) + 3(2) = 14

.

Step-by-Step Solution

1

Multiply matrix

A

and matrix

B

to express the product

AB

algebraically.

AB = \begin{bmatrix} x(2) + 3(y) & x(-1) + 3(5) \\ -2(2) + 4(y) & -2(-1) + 4(5) \end{bmatrix} = \begin{bmatrix} 2x + 3y & -x + 15 \\ -4 + 4y & 22 \end{bmatrix}

To solve for the variables, we must first determine the resulting matrix from the multiplication of

A

and

B

.

2

Set up equations by equating the entries of

AB

with the corresponding entries of matrix

C

.

2x + 3y = 14

,

-x + 15 = 11

, and

-4 + 4y = 4

Two matrices are equal if and only if all of their corresponding elements are equal.

3

Solve the single-variable equation from the first row, second column for

x

.

-x + 15 = 11 \implies -x = -4 \implies x = 4

Isolating

x

gives its numeric value.

4

Solve the single-variable equation from the second row, first column for

y

.

-4 + 4y = 4 \implies 4y = 8 \implies y = 2

Isolating

y

gives its numeric value.

5

Verify consistency with the remaining equation and calculate the final sum

x + y

.

2(4) + 3(2) = 8 + 6 = 14

(consistent), and

x + y = 4 + 2 = 6

.

Checking the first row, first column confirms the values are correct, and adding them yields the requested sum.

Key Concept

Matrix multiplication and matrix equality

Question 8Question →

A clothing boutique sells two types of shirts: t-shirts and polo shirts. The number of shirts sold at the boutique's two locations, Downtown and Uptown, on a weekend is represented by matrix

S

:

S = \begin{pmatrix} 40 & 30 \\ 50 & 20 \\end{pmatrix}

where the rows represent the locations (Downtown and Uptown, respectively) and the columns represent the shirt types (t-shirts and polo shirts, respectively).

The selling price and the production cost per shirt (in dollars) are represented by matrix

C

:

C = \begin{pmatrix} 15 & 6 \\ 25 & 10 \\end{pmatrix}

where the rows represent the shirt types (t-shirts and polo shirts, respectively) and the columns represent the selling price and production cost, respectively.

Which of the following matrices represents the total selling revenue and total production cost for the two locations?

\begin{pmatrix} 1350 & 540 \\ 1250 & 500 \\end{pmatrix}

\begin{pmatrix} 600 & 180 \\ 1250 & 200 \\end{pmatrix}

\begin{pmatrix} 900 & 570 \\ 1500 & 950 \\end{pmatrix}

\begin{pmatrix} 1850 & 740 \\ 950 & 380 \\end{pmatrix}

\begin{pmatrix} 780 & 1300 \\ 870 & 1450 \\end{pmatrix}

Show answer & explanation

Answer: \begin{pmatrix} 1350 & 540 \\ 1250 & 500 \\end{pmatrix}

Answer

The matrix representing the total selling revenue and total production cost for the two locations is the matrix with top row 1350 and 540, and bottom row 1250 and 500.

The correct matrix is obtained by multiplying the sales matrix

S

by the price/cost matrix

C

. Standard matrix multiplication pairs the quantity of each shirt type sold at each store with its respective selling price and production cost, yielding the correct total revenue and total production cost for both locations.

Step-by-Step Solution

1

Identify the dimensions and layout of the sales matrix

S

and the price/cost matrix

C

.

Matrix

S

is a

2 \times 2

matrix representing sales. Matrix

C

is a

2 \times 2

matrix representing price and cost. The product

SC

is defined because the number of columns in

S

(2) equals the number of rows in

C

(2).

Before multiplying, we must verify that the dimensions are compatible and that the product yields the desired real-world quantities.

2

Calculate the entries for the first row of the resulting matrix

SC

(Downtown location) by taking the dot product of the first row of

S

with the columns of

C

.

Revenue (Row 1, Column 1):

40 \times 15 + 30 \times 25 = 600 + 750 = 1350

.
Cost (Row 1, Column 2):

40 \times 6 + 30 \times 10 = 240 + 300 = 540

.

This yields the total revenue and production cost for the Downtown location by combining sales and prices/costs for both shirt types.

3

Calculate the entries for the second row of the resulting matrix

SC

(Uptown location) by taking the dot product of the second row of

S

with the columns of

C

.

Revenue (Row 2, Column 1):

50 \times 15 + 20 \times 25 = 750 + 500 = 1250

.
Cost (Row 2, Column 2):

50 \times 6 + 20 \times 10 = 300 + 200 = 500

.

This yields the total revenue and production cost for the Uptown location.

Key Concept

Matrix multiplication

Question 9Question →

Let matrix $X = \begin{bmatrix} 5 & -3 \\ 2 & 1 \end{bmatrix}$ and matrix $Y = \begin{bmatrix} 1 & 4 \\ -2 & 3 \end{bmatrix}$ . If $Z = X - Y$ , what is the value of the element in the first row and second column of $Z$ ?

Show answer & explanation

Answer: -7

Answer

The correct answer is

-7

.

To find the element in the first row and second column of matrix

Z = X - Y

, we subtract the element in the first row and second column of matrix

Y

from the corresponding element in matrix

X

. The element in the first row and second column of

X

is

-3

, and the element in the first row and second column of

Y

is

4

. Subtracting these values gives

-3 - 4 = -7

.

Step-by-Step Solution

1

Identify the elements in the first row and second column for both matrices.

x_{1,2} = -3

and

y_{1,2} = 4

To find the element in the first row and second column of the resulting matrix

Z

, we must use the corresponding elements from matrices

X

and

Y

.

2

Subtract the element of

Y

from the element of

X

.

-3 - 4 = -7

Since

Z = X - Y

, each element

z_{i,j}

of the resulting matrix is calculated as

x_{i,j} - y_{i,j}

.

Key Concept

Matrix Subtraction

Question 10Question →

A sporting goods company sells soccer balls and basketballs at two locations, North and South. The matrix $A$ represents the inventory of these balls at each location:

A = \begin{bmatrix} 50 & 30 \\ 40 & 60 \end{bmatrix}

where the rows represent the locations (Row 1: North, Row 2: South) and the columns represent the ball types (Column 1: soccer balls, Column 2: basketballs).

The matrix $B$ represents the price, in dollars, of each type of ball under two different pricing structures (Column 1: regular price, Column 2: sale price):

B = \begin{bmatrix} 20 & 15 \\ 25 & 18 \end{bmatrix}

where Row 1 represents soccer balls and Row 2 represents basketballs.

Which of the following matrices represents the total revenue, in dollars, the company would collect from selling its entire inventory at each location under both pricing structures?

\begin{bmatrix} 1,000 & 450 \\ 1,000 & 1,080 \end{bmatrix}

\begin{bmatrix} 1,600 & 1,500 \\ 1,970 & 1,830 \end{bmatrix}

\begin{bmatrix} 250 & 210 \\ 700 & 480 \end{bmatrix}

\begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}

\begin{bmatrix} 1,750 & 1,380 \\ 2,300 & 1,720 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}$

Answer

\begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}

To find the total revenue under both pricing structures at each location, we perform the matrix multiplication

AB

. The entry in row

i

, column

j

of the resulting matrix corresponds to the revenue generated at location

i

under pricing structure

j

. Applying the row-by-column multiplication rule gives the correct matrix containing

1,750

,

1,290

,

2,300

, and

1,680

.

Step-by-Step Solution

1

Set up the matrix multiplication to compute the total revenue.

The total revenue matrix is represented by the product

AB

, where we multiply the

2 \times 2

inventory matrix

A

by the

2 \times 2

price matrix

B

.

Multiplying the quantities of items by their respective unit prices yields the total revenue.

2

Calculate the entries for the first row of the product matrix, representing the North location.

Row 1, Column 1 (North, Regular):

50(20) + 30(25) = 1,000 + 750 = 1,750

.
Row 1, Column 2 (North, Sale):

50(15) + 30(18) = 750 + 540 = 1,290

.

The first row of matrix

A

is multiplied by the columns of matrix

B

to find revenues under both pricing structures at the North location.

3

Calculate the entries for the second row of the product matrix, representing the South location.

Row 2, Column 1 (South, Regular):

40(20) + 60(25) = 800 + 1,500 = 2,300

.
Row 2, Column 2 (South, Sale):

40(15) + 60(18) = 600 + 1,080 = 1,680

.

The second row of matrix

A

is multiplied by the columns of matrix

B

to find revenues under both pricing structures at the South location.

4

Construct the final product matrix from the calculated values.

AB = \begin{bmatrix} 1,750 & 1,290 \\ 2,300 & 1,680 \end{bmatrix}

Placing the computed row values in their respective positions yields the completed revenue matrix.

Key Concept

Matrix multiplication is used to combine quantity data and unit cost data systematically across multiple categories and scenarios.

Question 11Question →

A sporting goods chain has two locations. Location 1 sells 10 soccer balls and 15 basketballs. Location 2 sells 8 soccer balls and 20 basketballs. This is represented by the sales matrix $A = \begin{bmatrix} 10 & 15 \\ 8 & 20 \\end{bmatrix}$ . Soccer balls cost $15 each and basketballs cost$ 25 each, represented by the price matrix $B = \begin{bmatrix} 15 \\ 25 \\end{bmatrix}$ . What is the product $AB$ , which represents the total revenue generated by each location?

\begin{bmatrix} 150 \\ 500 \\end{bmatrix}

\begin{bmatrix} 350 \\ 725 \\end{bmatrix}

\begin{bmatrix} 525 \\ 520 \\end{bmatrix}

\begin{bmatrix} 525 \\ 620 \\end{bmatrix}

\begin{bmatrix} 525 & 620 \\end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 525 \\ 620 \\end{bmatrix}$

Answer

The column matrix with elements 525 in the first row and 620 in the second row

The correct answer is the column matrix containing 525 and 620. This is found by multiplying each row of the sales matrix (representing each location) by the column of the price matrix (representing prices of the items):

10(15) + 15(25) = 525

for the first row, and

8(15) + 20(25) = 620

for the second row.

Step-by-Step Solution

1

Set up the matrix multiplication of the

2 \times 2

sales matrix

A

and the

2 \times 1

price matrix

B

.

AB = \begin{bmatrix} 10 & 15 \\ 8 & 20 \\end{bmatrix} \begin{bmatrix} 15 \\ 25 \\end{bmatrix}

To find the total revenue for each location, we multiply the sales of each item by its corresponding price.

2

Multiply the first row of

A

by the column of

B

to find the element in the first row of the product matrix (revenue for Location 1).

10(15) + 15(25) = 150 + 375 = 525

This calculates the total revenue for Location 1 by summing the revenue from soccer balls (

10 \times 15

) and basketballs (

15 \times 25

).

3

Multiply the second row of

A

by the column of

B

to find the element in the second row of the product matrix (revenue for Location 2).

8(15) + 20(25) = 120 + 500 = 620

This calculates the total revenue for Location 2 by summing the revenue from soccer balls (

8 \times 15

) and basketballs (

20 \times 25

).

4

Combine the results into a

2 \times 1

resulting matrix.

\begin{bmatrix} 525 \\ 620 \\end{bmatrix}

Since matrix

A

has dimensions

2 \times 2

and matrix

B

has dimensions

2 \times 1

, their product

AB

must have dimensions

2 \times 1

.

Key Concept

Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second matrix, matching corresponding elements and adding the products.

Alternative Method

Instead of matrix multiplication, the revenue can be computed using standard arithmetic: Location 1 revenue is

10 \times \$15 + 15 \times \$25 = \$525

and Location 2 revenue is

8 \times \$15 + 20 \times \$25 = \$620

. The resulting column matrix is

\begin{bmatrix} 525 \\ 620 \\end{bmatrix}

.

Estimated Time:1m 0s

Question 12Question →

The matrices $M$ and $N$ are defined as follows:

M = \begin{bmatrix} 4 & -2 \\ 1 & 3 \end{bmatrix}, \quad N = \begin{bmatrix} 2 & 5 \\ 0 & -1 \end{bmatrix}

If the product matrix $P$ is defined by the equation $P = MN$ , what is the value of the element in the first row and second column of $P$ ?

Show answer & explanation

Answer: 22

Answer

The value of the element in the first row and second column of

P

is 22.

To find the element in the first row and second column of the product matrix

P = MN

, we calculate the dot product of the first row of matrix

M

(

[4, -2]

) and the second column of matrix

N

(

\begin{bmatrix} 5 \\ -1 \end{bmatrix}

):

4(5) + (-2)(-1) = 20 + 2 = 22

.

Step-by-Step Solution

1

Determine the row of the left matrix and the column of the right matrix needed for the target element.

To find the element in row 1, column 2 of

P = MN

, use the first row of

M

, which is

[4, -2]

, and the second column of

N

, which is

\begin{bmatrix} 5 \\ -1 \end{bmatrix}

.

The entry in row

i

and column

j

of a product matrix is the dot product of row

i

of the first matrix and column

j

of the second matrix.

2

Compute the dot product of the selected row and column.

(4 \times 5) + (-2 \times -1) = 20 + 2

Multiply corresponding elements and sum the products.

3

Simplify to get the final numeric answer.

22

Adding 20 and 2 yields 22.

Key Concept

Matrix multiplication involves multiplying the rows of the first matrix by the columns of the second matrix.

Question 13Question →

A catering service is organizing two events: a wedding and a corporate gala. The matrix $P$ represents the number of Deluxe packages ( $d$ ) and Premium packages ( $p$ ) ordered for each event:

P = \begin{bmatrix} 50 & 40 \\ 80 & x \end{bmatrix}

where the rows represent the Wedding and the Gala respectively, and the columns represent Deluxe and Premium packages respectively.

The matrix $R$ represents the requirements (in units) of protein and vegetables for each package type:

R = \begin{bmatrix} 3 & 2 \\ 4 & y \end{bmatrix}

where the rows represent Deluxe and Premium packages respectively, and the columns represent protein and vegetables respectively.

The total units of ingredients required for each event are represented by the product matrix $T = P \times R$ .

If the total vegetables required for the wedding is 220 units, and the total protein required for the corporate gala is 560 units, what is the value of the sum $x + y$ ?

11

54.5

82.8

83

145.5

Show answer & explanation

Answer: 83

Answer

83

The correct option is the value 83. Setting up the product matrix

T = P \times R

, the entry in row 1, column 2 (representing wedding vegetables) is

50(2) + 40y = 220

, which simplifies to

y = 3

. The entry in row 2, column 1 (representing gala protein) is

80(3) + 4x = 560

, which simplifies to

x = 80

. Adding these values together gives

x + y = 83

.

Step-by-Step Solution

1

Set up the matrix product expression

T = P \times R = \begin{bmatrix} 50 & 40 \\ 80 & x \end{bmatrix} \begin{bmatrix} 3 & 2 \\ 4 & y \end{bmatrix} = \begin{bmatrix} 50(3) + 40(4) & 50(2) + 40(y) \\ 80(3) + 4x & 80(2) + xy \end{bmatrix}

We multiply row elements of the first matrix by column elements of the second matrix to obtain the entries of the product matrix.

2

Isolate the equation for vegetables required for the wedding

50(2) + 40y = 220 \implies 100 + 40y = 220 \implies 40y = 120 \implies y = 3

The wedding corresponds to the first row of matrix

P

, and vegetables correspond to the second column of matrix

R

. Their dot product equals the total vegetables for the wedding.

3

Isolate the equation for protein required for the corporate gala

80(3) + 4x = 560 \implies 240 + 4x = 560 \implies 4x = 320 \implies x = 80

The corporate gala corresponds to the second row of matrix

P

, and protein corresponds to the first column of matrix

R

. Their dot product equals the total protein for the gala.

4

Find the sum of the variables

x + y = 80 + 3 = 83

We add the solved values of

x

and

y

to obtain the requested sum.

Key Concept

Matrix Multiplication and Word Problem Translation

Estimated Time:3m 0s

Question 14Question →

A construction company is working on two building projects, Project X and Project Y. The quantities of concrete (in tons) and steel (in tons) required for these projects are represented by the matrix $Q$ :

Q = \begin{bmatrix} 80 & 25 \\ 120 & 40 \end{bmatrix}

where the first row represents Project X, the second row represents Project Y, the first column represents concrete, and the second column represents steel.

The purchase cost per ton and the transportation cost per ton for these materials are represented by the matrix $C$ :

C = \begin{bmatrix} 110 & 15 \\ 650 & 45 \end{bmatrix}

where the first row represents concrete, the second row represents steel, the first column represents the purchase cost (in dollars per ton), and the second column represents the transportation cost (in dollars per ton).

If the product matrix $P = QC$ represents the total cost details for the two projects, what is the total transportation cost for Project Y, in dollars?

Show answer & explanation

Answer: 3600

Answer

The total transportation cost for Project Y is 3,600 dollars.

The correct answer is 3,600. To find the total transportation cost for Project Y, we calculate the entry in the second row (Project Y) and second column (transportation cost) of the product matrix

P = QC

. This is calculated as

120 \text{ tons of concrete} \times 15 \text{ dollars per ton} + 40 \text{ tons of steel} \times 45 \text{ dollars per ton} = 1,800 + 1,800 = 3,600

dollars.

Step-by-Step Solution

1

Identify the row representing Project Y and the column representing transportation cost

Row 2 of

Q

is

[120, 40]

and Column 2 of

C

is

[15, 45]^T

.

To find the total transportation cost for Project Y, we must compute the entry in the second row (Project Y) and second column (transportation cost) of the product matrix

P = QC

.

2

Multiply the row elements by the corresponding column elements

120 \times 15 = 1800

and

40 \times 45 = 1800

This calculates the individual transportation costs for the concrete and steel required for Project Y.

3

Sum the products to find the total transportation cost

1800 + 1800 = 3600

Adding these individual costs gives the total transportation cost for Project Y.

Key Concept

Matrix multiplication involves multiplying the elements of each row of the first matrix by the corresponding elements of each column of the second matrix and summing the products.

Question 15Question →

A local farm sells organic fruit baskets at a weekend market. The number of small and large baskets sold on Saturday and Sunday is represented by matrix $Q$ , where row 1 represents Saturday, row 2 represents Sunday, column 1 represents small baskets, and column 2 represents large baskets:

Q = \begin{bmatrix} 40 & 30 \\ 20 & 50 \end{bmatrix}

The selling price and the production cost, in dollars, for each type of basket are represented by matrix $P$ , where row 1 represents small baskets, row 2 represents large baskets, column 1 represents the selling price, and column 2 represents the production cost:

P = \begin{bmatrix} 15 & 8 \\ 25 & 12 \end{bmatrix}

The product matrix $R = QP$ represents the total revenue and total production cost for each day. Which of the following matrices represents $R$ ?

\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}

\begin{bmatrix} 600 & 240 \\ 500 & 600 \end{bmatrix}

\begin{bmatrix} 760 & 850 \\ 1240 & 1350 \end{bmatrix}

\begin{bmatrix} 1350 & 1550 \\ 680 & 760 \end{bmatrix}

\begin{bmatrix} 1450 & 680 \\ 1550 & 660 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}$

Answer

\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}

The correct product matrix is obtained by performing matrix multiplication

QP

. Multiplying the first row of

Q

by the first column of

P

yields the Saturday revenue:

40 \times 15 + 30 \times 25 = 1350

. Multiplying the first row of

Q

by the second column of

P

yields the Saturday cost:

40 \times 8 + 30 \times 12 = 680

. Repeating this for the second row of

Q

(Sunday) yields the Sunday revenue of

1550

and Sunday cost of

760

. Thus, the resulting matrix is the one with row 1 equal to

[1350, 680]

and row 2 equal to

[1550, 760]

.

Step-by-Step Solution

1

Set up the matrix multiplication product

R = QP

.

R = \begin{bmatrix} 40 & 30 \\ 20 & 50 \end{bmatrix} \begin{bmatrix} 15 & 8 \\ 25 & 12 \end{bmatrix}

To find the total revenue and production cost for Saturday and Sunday, we must multiply the quantity matrix by the price-cost matrix.

2

Calculate the elements of the first row of the product matrix

R

, representing Saturday's revenue and cost.

Row 1, Column 1 (Saturday Revenue):

40(15) + 30(25) = 600 + 750 = 1350

. Row 1, Column 2 (Saturday Cost):

40(8) + 30(12) = 320 + 360 = 680

.

Multiply the first row of matrix

Q

by the columns of matrix

P

to determine Saturday's financial values.

3

Calculate the elements of the second row of the product matrix

R

, representing Sunday's revenue and cost.

Row 2, Column 1 (Sunday Revenue):

20(15) + 50(25) = 300 + 1250 = 1550

. Row 2, Column 2 (Sunday Cost):

20(8) + 50(12) = 160 + 600 = 760

.

Multiply the second row of matrix

Q

by the columns of matrix

P

to determine Sunday's financial values.

4

Combine the calculated elements into the final

2 \times 2

matrix.

\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}

Placing the values in their corresponding row and column positions yields the complete product matrix.

Key Concept

Matrix Multiplication in Applied Word Problems

Estimated Time:1m 30s

Question 16Question →

A company sells two types of gift baskets: Standard and Premium. The sales at the company's North and South branches are represented by matrix $S$ :

S = \begin{bmatrix} 12 & 8 \\ 15 & 10 \end{bmatrix}

where the rows represent the North and South branches, respectively, and the columns represent the number of Standard and Premium baskets sold, respectively. The cost of the items inside each basket is represented by matrix $C$ :

C = \begin{bmatrix} x & 24 \\ 30 & y \end{bmatrix}

where the rows represent Standard and Premium baskets, respectively, and the columns represent the cost of food items and the cost of gift items (in dollars), respectively. If the total food item cost at the North branch is $600$ and the total gift item cost at the South branch is $760$ , what is the value of $x + y$ ?

Show answer & explanation

Answer: 70

Answer

The value of

x + y

is 70.

To find the total costs, the sales matrix

S

is multiplied by the unit cost matrix

C

. The product matrix

R = SC

is a

2 \times 2

matrix where the entry in Row 1, Column 1 represents the total food cost at the North branch, and the entry in Row 2, Column 2 represents the total gift cost at the South branch. Setting up the equations

12x + 240 = 600

and

360 + 10y = 760

yields

x = 30

and

y = 40

, which sum to 70.

Step-by-Step Solution

1

Set up the matrix multiplication

R = SC

to represent the total costs.

R = \begin{bmatrix} 12x + 240 & 288 + 8y \\ 15x + 300 & 360 + 10y \end{bmatrix}

The product of the sales matrix and the unit cost matrix yields the total cost matrix, where rows represent the branches and columns represent the cost categories.

2

Identify the expression for the total food item cost at the North branch and solve for

x

.

x = 30

The total food item cost at the North branch is the entry in Row 1, Column 1 of the product matrix, which is

12x + 8(30) = 12x + 240

. Setting this equal to

600

gives

12x + 240 = 600

, which simplifies to

12x = 360

, so

x = 30

.

3

Identify the expression for the total gift item cost at the South branch and solve for

y

.

y = 40

The total gift item cost at the South branch is the entry in Row 2, Column 2 of the product matrix, which is

15(24) + 10y = 360 + 10y

. Setting this equal to

760

gives

360 + 10y = 760

, which simplifies to

10y = 400

, so

y = 40

.

4

Calculate the sum of the variables

x

and

y

.

x + y = 70

Substitute

x = 30

and

y = 40

into the expression

x + y

to find the final answer.

Key Concept

Matrix multiplication and translating real-world scenarios into matrix equations.

Estimated Time:3m 0s

Question 17Question →

Matrices $A$ and $B$ are defined as:

A = \begin{bmatrix} x & 4 \\ -3 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -2 \\ 5 & 3 \end{bmatrix}

Let $C$ represent the product matrix $AB$ , where:

C = \begin{bmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{bmatrix}

If the element $c_{12}$ is equal to $2$ , what is the value of $x$ ?

Show answer & explanation

Answer: 5

Answer

5

To find the entry

c_{12}

in the first row and second column of the product matrix

C = AB

, we calculate the dot product of the first row of matrix

A

and the second column of matrix

B

. The first row of

A

is

[x, 4]

and the second column of

B

is

[-2, 3]^T

. Thus,

c_{12} = x(-2) + 4(3) = -2x + 12

. Setting this equal to the given value of

2

gives the equation

-2x + 12 = 2

. Subtracting

12

from both sides results in

-2x = -10

. Dividing by

-2

yields

x = 5

.

Step-by-Step Solution

1

Identify the formula for the element

c_{12}

in the product matrix

C = AB

.

c_{12} = -2x + 12

The element

c_{12}

is located in the first row and second column of the product matrix, so it is the product of the first row of

A

, which is

[x, 4]

, and the second column of

B

, which is

[-2, 3]^T

.

2

Set the expression for

c_{12}

equal to the given value of

2

and solve the linear equation for

x

.

x = 5

Setting

-2x + 12 = 2

leads to

-2x = -10

, and dividing both sides by

-2

yields

x = 5

.

Key Concept

Matrix Multiplication and Element-wise Operations

Question 18Question →

A community theater group sells student tickets and adult tickets for their weekend plays. The number of tickets sold for Friday and Saturday night shows is represented by the matrix $T$ :

T = \begin{bmatrix} 80 & 120 \\ 100 & 150 \end{bmatrix}

where the first and second rows represent Friday and Saturday, respectively, and the first and second columns represent student and adult tickets, respectively. The price of each ticket is represented by the matrix $P$ :

P = \begin{bmatrix} 6 \\ 10 \end{bmatrix}

where the first row represents the price of a student ticket ( $6) and the second row represents the price of an adult ticket ($ 10). Which of the following matrices represents the total ticket sales revenue, in dollars, for Friday and Saturday, respectively?

\begin{bmatrix} 480 & 1,200 \\ 600 & 1,500 \end{bmatrix}

\begin{bmatrix} 2,100 \\ 1,680 \end{bmatrix}

\begin{bmatrix} 1,680 \\ 2,100 \end{bmatrix}

\begin{bmatrix} 1,580 \\ 2,100 \end{bmatrix}

\begin{bmatrix} 1,520 \\ 1,900 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 1,680 \\ 2,100 \end{bmatrix}$

Answer

The matrix with entries 1,680 and 2,100

The correct answer is the matrix with entries 1,680 and 2,100. This is found by multiplying the sales matrix by the price matrix. The revenue for Friday is calculated as

80 \times 6 + 120 \times 10 = 480 + 1,200 = 1,680

. The revenue for Saturday is

100 \times 6 + 150 \times 10 = 600 + 1,500 = 2,100

. Placing these in a column matrix matching the order of the days yields the correct matrix.

Step-by-Step Solution

1

Set up the matrix multiplication of the ticket sales matrix

T

and the ticket price matrix

P

.

TP = \begin{bmatrix} 80 & 120 \\ 100 & 150 \end{bmatrix} \begin{bmatrix} 6 \\ 10 \end{bmatrix}

To find the total revenue for Friday and Saturday, we must multiply the ticket sales by their respective prices.

2

Perform the multiplication by taking the dot product of each row of

T

with the column of

P

.

\begin{bmatrix} 80(6) + 120(10) \\ 100(6) + 150(10) \end{bmatrix} = \begin{bmatrix} 480 + 1200 \\ 600 + 1500 \end{bmatrix}

The matrix product of a

2 \times 2

matrix and a

2 \times 1

matrix results in a

2 \times 1

matrix where the entries are the sums of the products of corresponding elements.

3

Simplify the sums in the resulting matrix.

\begin{bmatrix} 1,680 \\ 2,100 \end{bmatrix}

Adding the products gives the final total revenues for Friday and Saturday, respectively.

Key Concept

Matrix Multiplication in Word Problems

Alternative Method

Instead of formal matrix multiplication, you can calculate the scalar totals for each day directly (Friday:

80 \times 6 + 120 \times 10 = 1,680

; Saturday:

100 \times 6 + 150 \times 10 = 2,100

) and match them with the corresponding rows of the resulting

2 \times 1

matrix.

Estimated Time:1m 30s

Question 19Question →

A cleaning service company offers two types of packages: Standard Clean and Deep Clean. The number of packages sold during the first two weeks of the month is represented by matrix $Q$ :

Q = \begin{bmatrix} 20 & 15 \\ 25 & 18 \end{bmatrix}

where row 1 represents Week 1, row 2 represents Week 2, column 1 represents Standard Clean packages, and column 2 represents Deep Clean packages.

The price charged and the cost to perform each package (in dollars) are represented by matrix $P$ :

P = \begin{bmatrix} 80 & 30 \\ 150 & 60 \end{bmatrix}

where row 1 represents Standard Clean packages, row 2 represents Deep Clean packages, column 1 represents the price charged, and column 2 represents the cost.

Which of the following matrices represents the total revenue and total cost for Week 1 and Week 2, respectively?

\begin{bmatrix} 1,600 & 450 \\ 3,750 & 1,080 \end{bmatrix}

\begin{bmatrix} 2,350 & 1,740 \\ 4,500 & 3,330 \end{bmatrix}

\begin{bmatrix} 3,850 & 1,500 \\ 4,700 & 1,830 \end{bmatrix}

\begin{bmatrix} 3,800 & 1,500 \\ 4,000 & 1,830 \end{bmatrix}

\begin{bmatrix} 650 & 300 \\ 700 & 330 \end{bmatrix}

Show answer & explanation

Answer: $\begin{bmatrix} 3,850 & 1,500 \\ 4,700 & 1,830 \end{bmatrix}$

Answer

The matrix with elements 3,850 and 1,500 in the first row, and 4,700 and 1,830 in the second row.

The matrix representing the total revenue and total cost for Week 1 and Week 2 is found by multiplying matrix

Q

by matrix

P

. Since

Q

is a

2 \times 2

matrix (weeks by package types) and

P

is a

2 \times 2

matrix (package types by financial categories), the product

Q \times P

yields a

2 \times 2

matrix representing weeks by financial categories (revenue and cost).

Calculating each element:
- Row 1, Column 1 (Week 1 Revenue):

(20 \times 80) + (15 \times 150) = 1,600 + 2,250 = 3,850

- Row 1, Column 2 (Week 1 Cost):

(20 \times 30) + (15 \times 60) = 600 + 900 = 1,500

- Row 2, Column 1 (Week 2 Revenue):

(25 \times 80) + (18 \times 150) = 2,000 + 2,700 = 4,700

- Row 2, Column 2 (Week 2 Cost):

(25 \times 30) + (18 \times 60) = 750 + 1,080 = 1,830

This results in the matrix:

\begin{bmatrix} 3,850 & 1,500 \\ 4,700 & 1,830 \end{bmatrix}

Step-by-Step Solution

1

Identify the appropriate matrix operation to combine the weekly package sales with their respective price and cost.

We need to compute the product of matrix

Q

and matrix

P

(

Q \times P

), since multiplying a

2 \times 2

matrix of (weeks

\times

packages) by a

2 \times 2

matrix of (packages

\times

financial category) results in a

2 \times 2

matrix of (weeks

\times

financial category).

This alignment ensures that the package types cancel out, leaving the desired weekly revenue and cost breakdown.

2

Calculate the elements of the first row of the product matrix.

Row 1, Column 1 (Week 1 Revenue):

20(80) + 15(150) = 1,600 + 2,250 = 3,850

. Row 1, Column 2 (Week 1 Cost):

20(30) + 15(60) = 600 + 900 = 1,500

.

The first row represents Week 1 revenue and cost.

3

Calculate the elements of the second row of the product matrix.

Row 2, Column 1 (Week 2 Revenue):

25(80) + 18(150) = 2,000 + 2,700 = 4,700

. Row 2, Column 2 (Week 2 Cost):

25(30) + 18(60) = 750 + 1,080 = 1,830

.

The second row represents Week 2 revenue and cost.

Key Concept

Matrix multiplication represents linear combinations of variables, useful for calculating total cost and revenue in multi-variable business scenarios.

Question 20Question →

Let matrix $A = \begin{bmatrix} 3 & -2 \\ 5 & 1 \end{bmatrix}$ and matrix $B = \begin{bmatrix} -2 & 3 \\ 2 & -3 \end{bmatrix}$ . If matrix $C$ is defined by the equation $C = 3A - 2B$ , what is the value of the element in the second row and first column of $C$ ?

Show answer & explanation

Answer: 11

Answer

The element in the second row and first column of matrix

C

is 11.

To find the element in the second row and first column of matrix

C

, we apply the operations defined by

C = 3A - 2B

directly to the corresponding elements of

A

and

B

. The element in row 2, column 1 of matrix

A

is

5

, and of matrix

B

is

2

. Computing

3(5) - 2(2)

yields

15 - 4 = 11

.

Step-by-Step Solution

1

Identify the elements in the second row and first column of matrices

A

and

B

.

A_{21} = 5

and

B_{21} = 2

.

To find a specific element of the resulting matrix

C = 3A - 2B

, we only need to perform the operations on the elements in the corresponding position.

2

Set up the equation for the element in the second row and first column of

C

.

C_{21} = 3A_{21} - 2B_{21}

Matrix addition, subtraction, and scalar multiplication are performed element-wise.

3

Substitute the identified values into the equation and compute the result.

C_{21} = 3(5) - 2(2) = 15 - 4 = 11

Evaluating the expression gives the value of the target element.

Key Concept

Matrix scalar multiplication and element-wise subtraction

Matrices and Matrix Operations

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes