Algebra 2

1
Algebra 2

• Section 4-3
• Multiplying Matrices

2
What You'll LearnWhy It's Important

• To multiply matrices
• You can use matrices to solve problems involving
  probability and track and field

3
Application Sales

• The manager of DK's Donuts makes a daily report
  to the owner that summarizes the cost of each
  kind of donut and the number of donuts sold for
  that day. The sales for one day are summarized in
  the cost matrix C and sales matrix S.

4
Application Sales

• You can use matrix multiplication to find the
  income for the day. In this case, multiply each
  element in the cost matrix by its corresponding
  element in the sales matrix and find the total.

5
Application Sales
The income for the day was 260.70

• Each element in the row matrix is multiplied by
  an element in the column matrix

6
Multiplying Matrices

• In general, the product of two matrices is found
  by multiplying rows and columns.
• The product of Am x n and Bn x r is (AB)m x r.
  The element in the ith row and the jth column of
  AB is the sum of the products of the
  corresponding elements in the ith row of A and
  the jth column of B.

7
When can I multiply Matrices?

• You can multiply two matrices only if the number
  of columns in the first matrix is equal to the
  number of rows in the second matrix.

2 x 2
2 x 3
2 x 3
2 x 2
notpossible
possible

• Remember dimensions are given in row x column

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Example 1
9
Solution Example 1

• Can these two matrices be multiplied?
• Does the number of columns in the first matrix
  equal the number of rows in the second matrix?
• YES
• What are the dimensions of the matrix product
  (answer)?

A2 x 2? B2 x 3
2 x 3
10
Solution Example 1
5
1
-3
3
-5
2
7
8
-4
9

• Multiply 1st row x 1st column

11
Solution Example 1
5
1
-3
3
-5
2
7
8
-4
9

• Multiply 1st row x 1st column
• Multiply corresponding positions

12
Solution Example 1
5
1
-3
3
-5
2
7
8
-4
9

• Multiply 1st row x 1st column
• Multiply corresponding positions and add

13
Solution Example 1
5
1
-3
3
-5
2
7
8
-4
9

• Multiply 1st row x 2nd column
• Multiply corresponding positions

14
Solution Example 1
5
1
-3
3
-5
2
7
8
-4
9

• Multiply 1st row x 2nd column
• Multiply corresponding positions and add

15
Solution Example 1
5
1
-3
3
-5
2
7
8
-4
9

• Multiply 1st row x 3rd column
• Multiply corresponding positions

16
Solution Example 1
5
1
-3
3
-5
2
7
8
-4
9

• Multiply 1st row x 3rd column
• Multiply corresponding positions and add

17
Solution Example 1
5
1
-3
3
-5
2
7
8
-4
9

• Multiply 2nd row x 1st column
• Multiply corresponding positions

18
Solution Example 1
5
1
-3
3
-5
2
7
8
-4
9

• Multiply 2nd row x 1st column
• Multiply corresponding positions and add

19
Solution Example 1
5
1
-3
3
-5
2
7
8
-4
9

• Multiply 2nd row x 2nd column
• Multiply corresponding positions

20
Solution Example 1
5
1
-3
3
-5
2
7
8
-4
9

• Multiply 2nd row x 2nd column
• Multiply corresponding positions and add

21
Solution Example 1
5
1
-3
3
-5
2
7
8
-4
9

• Multiply 2nd row x 3rd column
• Multiply corresponding positions

22
Solution Example 1
5
1
-3
3
-5
2
7
8
-4
9

• Multiply 2nd row x 3rd column
• Multiply corresponding positions and add

23
Recap
1st row 1st column
1st row 2nd column
1st row 3rd column
2nd row 1st column
2nd row 2nd column
2nd row 3rd column
24
Solution Example 1
25
Geometry Transformations

• Another use of matrix multiplication is in
  transformational geometry
• You have already learned about dilations and
  translations, another type of transformation is a
  rotation.
• A rotation occurs when a figure is moved around a
  center point.
• To move a figure by rotation, you can use a
  rotation matrix.

26
Rotation Matrix

• The matrix will rotate a figure
  onthe coordinate plane about the origin 90
  counterclockwise
• (see page 201 activity for proof of this
  statement)

27
Example 2

• Line AB passes through points A(4,-2) and
  B(-3,5). Find the coordinates of two points on
  line A'B' that has been rotated 90
  counterclockwise about the origin. Draw its graph
  and describe the relationship between lines AB
  and A'B'.

28
Solution Example 2

• Line AB passes through points A(4,-2) and
  B(-3,5). Find the coordinates of two points on
  line A'B' that has been rotated 90
  counterclockwise about the origin. Draw its graph
  and describe the relationship between lines AB
  and A'B'.

• Write the ordered pairs in a coordinate matrix.
  Then multiply the coordinate matrix by the
  rotation matrix.

?
29
Solution Example 2
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30
Solution Example 2
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31
Solution Example 2
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Solution Example 2
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Solution Example 2
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34
Solution Example 2

• Line AB passes through points A(4,-2) and
  B(-3,5). Find the coordinates of two points on
  line A'B' that has been rotated 90
  counterclockwise about the origin. Draw its graph
  and describe the relationship between lines AB
  and A'B'.

The two lines appear to be perpendicular
A'(2,4)

• The coordinates of the two points on the line are
  A'(2,4) and B'(-5,-3).

B'(-5,-3)
35
Video of Rotation
36

• THE END