Download Lesson 1.8 - Coweta County Schools

• How do I perform
basic matrix
operations?
1.8
Perform Basic Matrix Operations
Adding and Subtracting Matrices
To add or subtract two matrices, simply add or subtract
corresponding elements.
_______________
You can add or subtract matrices only if they have the
dimensions
same ____________.
a
 c
Subtracting a
Matrices
 c
Adding
Matrices
b    e f   a  e b  f 
d   g h  c  g d  h 
b    e f   a  e b  f 
d   g h  c  g d  h 
1.8
Perform Basic Matrix Operations
Example 1 Add and subtract matrices
Perform the indicated operation, if possible.

2
1
1

4




6

2
3




a.

b.  0 3   2 9
3 2
 1
 7 6  1 5
Solution
6  2 
2 X 2 and the
a. The dimensions of
are ________
 1
3
3

 are _______.
2 X 1 So, you _________
cannot
dimensions of
 2
subtract the matrices.
Perform Basic Matrix Operations
1.8
Example 1 Add and subtract matrices
Perform the indicated operation, if possible.

2
1
1

4




6

2
3




a.

b.  0 3   2 9
3 2
 1
 7 6  1 5
Solution
the same So, add the matrices.
b. The dimensions are __________.
 2
 0
 7
1  1  4   2  1 1   4 
3   2 9   0   2
39 
6  1 5  __________
6  5 
7  1 _______
  1  3
   2 12 
 ________

8
11


1.8
Perform Basic Matrix Operations
Example 2 Multiply a matrix by a scalar
Perform the indicated operation, if possible.
0

a. 3
  1
1

b.  2
3
2  7   30 32 3 7 
6 4 3__________
34
 1 36 ____



21
0
6



3
18
__________
_
12


0    21  20 
5
 5  
23  2__
__________



2
0

6 10
______
1.8
Perform Basic Matrix Operations
Checkpoint. Perform the indicated operation, if possible.


6
2
3
0





1.

 1
  1  5  7 
2 2
2 2
 9 2 
 8  4 
1.8
Perform Basic Matrix Operations
Checkpoint. Perform the indicated operation, if possible.
 1
2.  3
 2
3 2
9 6
4

0
 3  9 6  12
 6  27 18 0


1.8
Perform Basic Matrix Operations
Properties of Matrix Operations
Let A, B, and C be matrices with the same dimensions,
and let k be a scalar.
Associative Property of Addition
 A  B  C  __________
A  B  C 
Commutative Property of Addition
A  B  ______
B A
Distributive Property of Addition
k  A  B  __________
kA kB
Distributive Property of Subtraction
kA kB
k  A  B  __________
1.8
Perform Basic Matrix Operations
Example 3 Solve a multi-step problem
Pet Stores Two pet stores sell both dogs and
cats. Sales from each store for last month and
this month are shown below.
Last
Store 1 sold 42 dogs, 33 cats
Month: Store 2 sold 56 dogs, 21 cats
This
Store 1 sold 36 dogs, 51 cats
Month: Store 2 sold 48 dogs, 37 cats
Organize
Solutionthe data into matrices, then write and interpret a matrix
giving
the average
monthly
sales2 for
two month period.
1. Organize
the data
into two
x 2 the
matrices.
Last month (A)
dogs
cats
 42
33 

Store 2  _________
 56 21 
Store 1
This month (B)
dogs
cats
 36 51 

Store 2  _________
37
 48

Store 1
1.8
Perform Basic Matrix Operations
Example 3 Solve a multi-step problem
Pet Stores Two pet stores sell both dogs and
cats. Sales from each store for last month and
this month are shown below.
Last
Store 1 sold 42 dogs, 33 cats
Month: Store 2 sold 56 dogs, 21 cats
This
Store 1 sold 36 dogs, 51 cats
Month: Store 2 sold 48 dogs, 37 cats
Write a matrix for the average monthly sales by adding A and B and
then

1 multiplying 1the result by ½.  
36
33
42
51


A  B   



2
2   _______
56 21   _______
48 37  
1  78 84   39 42 

 


58
_______   _______
52 29 
2 104
1.8
Perform Basic Matrix Operations
Example 3 Solve a multi-step problem
Pet Stores Two pet stores sell both dogs and
cats. Sales from each store for last month and
this month are shown below.
Last
Store 1 sold 42 dogs, 33 cats
Month: Store 2 sold 56 dogs, 21 cats
This
Store 1 sold 36 dogs, 51 cats
Month: Store 2 sold 48 dogs, 37 cats
Interpret the matrix from Step 2. Store 1 sold an average of
42 cats, while Store 2 sold an average of ___
39 dogs and ___
52
____
29 cats.
dogs and ___
1.8
Perform Basic Matrix Operations
Example 4 Solve a matrix equation
Solve the matrix equation for x and y.
 3x 1  1  3   8  8

4  





0
6
2

2
y
8
0
 
 


Solution
Simplify the left side of the equation.
 3x 1  1  3   8  8

4  





  0 6  2  2 y    8 0
  3x  1
   8  8

2

4  



2
_________
_____
8
0
6  2 y  


 12 x  4
   8  8

8




 _________

 8 y   8 0
8 24_____

Write original
equation.
Add matrices inside
parentheses.
Perform scalar
multiplication.
1.8
Perform Basic Matrix Operations
Example 4 Solve a matrix equation
Solve the matrix equation for x and y.
 3x 1  1  3   8  8

4  





0
6
2

2
y
8
0
 
 


Solution
Equate corresponding elements and solve the two resulting
equations.
_______
12x  4  8
____
12
12 x ___
x  ___
1
24  8 y  0
_______
____
24
 8 y ___
y  ___
3
The solution is x = ____
3
 1 and y = ____.
1.8
Perform Basic Matrix Operations
Checkpoint. Complete the following exercise.
3. Solve the matrix equation for x and y.
 3  2 x  1 
 6   4 12


 2 









1
7

5
y
4   8  6






2  2 x  6  4
 2



3
5
y

1

  8

 4 4 x 12  4

 10 y  2

 6  8

12

 6
12

 6
1.8
Perform Basic Matrix Operations
Checkpoint. Complete the following exercise.
3. Solve the matrix equation for x and y.

 4 4 x 12  4 12

 10 y  2


 6  8  6

 10 y  2  8
4 x 12  12
4 x  24
x6
 10 y  10
y  1
1.8
Perform Basic Matrix Operations
Pg. 45, 1.8 #1-23