Algebra 1 > Notes > Yorkcounty Final > Yc > Lesson_30

Lesson 30 Matrices Applied

Lesson Goals: VA SOL Goals

A. 4 The student will use matrices to organize and manipulate data including matrix addition, subtraction, and scalar multiplication. Data will arise from business, industrial, and consumer situations.

This lesson contains 21 slides.

A matrix is a rectangular array of numbers. Each number in the matrix is called an element.

5 1 L M 3 2 N

0 −7

O P Q

The matrix to the left has two rows and three columns, thus its dimensions are 2x3.

If we were to label the rows and columns they would look like this: C1

R1 R2

C2

L M 3 2 N

5 1

C3

O P −7 Q 0

If you were asked what element is in R1C3 you should say 0, because it is in row 1 and column 3 of the matrix.

Try This: Answer the questions about the following matrix.

5 L M 1 M M −4 N

O P P 8 P Q

−6 0 2 −10 3

1) What are the dimensions of the matrix? 2) What element is in Row 3 and Column 2? R3C2?

Advance to the next slide to check your answers when finished.

Try This Solution: 1) 3x3 because there are 3 rows and 3 columns 2) 3

Matrix Additon/Subtraction: When adding or subtracting two matrices, simply add/subtract the elements in corresponding positions:

This means that unless two matrices have the exact same dimensions, they cannot be added or subtracted.

In this case you would simply write : not possible or no solution

Example 1: Find the sum of the following matrices:

−2 3 OL −1 2 O L +M M P P 1 − 4 4 − 3 N QN Q a)

c)

−3 5 O L M P N5 7Q −1 1 O L M P 3 − 1 N Q

b)

d)

−3 5 O L M P 5 − 7 N Q −1 5 O L M P 5 − 1 N Q

Example 1 Solution:

−2 3 OL −1 2 O L +M M P P 1 − 4 4 − 3 N QN Q To add the matrices we must add together corresponding elements: ( −2 + −1) L M N(1 + 4)

O ( −4 + −3) P Q ( 3 + 2)

Hence, the best answer choice is B).

−3 5 O L =M N5 −7P Q

Example 2: Find the difference of the following matrices: 12 0.3OL7 L −M M P −5 −4 QN 24 N

a)

c)

L M N

O P Q

19 2.3 −29 −3.3

5 L M −19 N

O P Q

2.3 −3.3

−2 −0.7

b)

d)

O P Q 5 L M −29 N 5 L M −29 N

O P Q

2.3 −3.3

O P Q

−2.3 3.3

Example 3: Solution 12 0.3OL7 L −M M P −5 −4 QN 24 N

−2 −0.7

O P Q

Subtract the numbers in corresponding positions.

12 − 7g b 0.3 − −2g L O b M P b−5 − 24gb−4 − −0.7gQ N 5 L M −29 N

The best answer is choice B).

O P Q

2.3 −3.3

Try This: Complete each of the following matrix addition/subtraction problems below. 1)

3)

−1OL3 O L − MP M P −6Q N9 QN

2)

6 3OL 1 −2 5O L +M M P P − 4 3 4 11 0 N QN Q

6 3OL1 −1O L +M M P P − 4 3 − 2 4 N QN Q

Advance to the next slide when you are ready to check your answers.

Try This Solution:

1)

−4 O L M P 15 Q N

2)

7 2O L M P −6 7 Q N

3) No solution, the matrices do not have equal dimensions, so they cannot be added.

Scalar Multiplication: A scalar is a number that is multiplied by a matrix. Like:

7 2O L −3M P −6 7 Q N

You simply distribute and multiply the scalar by each element inside the matrix.

−3 ⋅ 7g b −3 ⋅ 2g L O b M P b−3⋅ −6gb−3⋅ 7gQ N

−21 L =M N18

O P Q

−6 −21

Try This: Perform the following scalar matrix multiplication:

If A =

−1 2 L M 19 1 M M N6 0

O P P P Q

0 4 −3

.

Find 2A

Advance to the next slide to check you answer when finished.

Try This Solution:

−2 L M 38 M M 12 N

4 2 0

O P P −6P Q 0 8

Example 4: Application Problem Pat has two coffee shops. The income and expenses for three months for each store is provided in the matrices below. Find the monthly profits for each store.

Store 1 Store 2

Income

Expenses

100’s of Dollars

100’s of Dollars

Jul Aug Sept

Jul Aug Sept

141 L M 137 N

O 203P Q

165 183 158

L M 129 Store 2 N Store 1 137

Solution: The Profit made by a company is equal to the income made minus the expenses that need to be paid. Thus, this is a simple matrix subtraction problem.

O 187 P Q

159 172 153

Example 4 Solution Continued….

Income

141 L M 137 N

O 203P Q

165 183 158

-

Expenses

137 L M 129 N

O 187 P Q

159 172 153

141 − 137gb 165 − 159gb 183 − 172g L O b M P b137 − 129gb158 − 153gb203 − 187gQ N

4 6 11O L M P 8 5 16Q N

Now You Try It: The matrices below give the data of shorts and tops in stock sold during one month. What is the inventory for next month, assuming no shipments come in?

In Stock Sm

215 L M 200 Shorts N Tops

Med

Sold Lg

Sm Med Lg

153 Tops L O M P 165 Q Shorts N

350 175 300 150

Advance to check your answer when done.

O P Q

245 143 197 112

Try This Solution:

The amount of tops and shorts left for next month will equal the inventory for this month minus the number of items sold this month.

215 L M 200 N

O P Q

350 175 300 150

153 L M 165 N

O P Q

245 143 197 112

62 105 32O L M P 35 103 38Q N

Summary Questions: Turn these in to your teacher and do not begin the assignment on the next slide until you have received a response. 1) What is a matrix 2) How do you add or subtract two matrices which are the exact same dimensions? 3) Show an example of your answer to number 2. 4) How do you add or subtract two matrices of different dimensions? 5) Make up a problem that demonstrates scalar multiplication and solve it.