Download 3 - Mira Costa High School

3.5 Perform Basic Matrix Operations
Goal  Perform operations with matrices.
Your Notes
VOCABULARY
Matrix
A rectangular arrangement of numbers in rows and columns
Dimensions
The dimensions of a matrix with m rows and n columns are m  n.
Elements
The numbers in a matrix
Equal matrices
Matrices that have the same dimensions and equal elements in corresponding positions
Scalar
A real number by which you multiply a matrix
Scalar multiplication
The process of multiplying each element in a matrix by a scalar
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 same _dimensions_.
Adding
Matrices
a b 
e
c d    g



f  a  e b  f 

h  c  g d  h 
Subtracting
Matrices
a b   e
c d    g

 
f  a  e b  f 

h  c  g d  h 
Your Notes
Example 1
Add and subtract matrices
Perform the indicated operation, if possible.
 2 1  1  4
6  2  3


 
a. 

9
3 2 b. 0 3   2
1
 7 6  1
5
Solution
a. The dimensions of 6 2 are _2  2_ and the dimensions of 3
2
1 3
are _2  1_. So, you _cannot_ subtract the matrices.
b. The dimensions are _the same . So, add the matrices.
 2 1
 1  4   2  1 1  ( 4) 
 0 3   2
9    0  ( 2)
3  9



 7 6
 1
5  
7 1
6  5
1 3
 2 12
 8 11
Example 2
Multiply a matrix by a scalar
Perform the indicated operation, if possible.
a. 3 0 2  7    3(0) 3(2) 3( 7)
 1 6 4  3(1) 3(6) 3(4) 
0 6  21
 
12
  3 18
1 0    2(1)
 2(0)   2 0
b..  2 
 

  

3  5  2(3)  2( 5)   6 10 
Checkpoint Perform the indicated operation, if possible.
2  3
 6
1.   1  5  7

 

9
2




 8  4
 1 3  2
2.  3 2 9  6
 3 9 6
 6  27 18

0
 1
4
0
 12
0 
Your Notes
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
Distributive Property of Addition
Distributive Property of Subtraction
A  B  _B  A_
k(A  B) = _kA  kB_
k(A  6) = _kA  kB_
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 Month: Store 1 sold 42 dogs, 33 cats.
Store 2 sold 56 dogs, 21 cats.
This Month: Store 1 sold 36 dogs, 51 cats.
Store 2 sold 48 dogs, 37 cats.
Organize the data into matrices, then write and interpret a matrix giving the average
monthly sales for the two month period.
Organize the data into two 2  2 matrices.
Last month (A)
dogs cats
Store 1 42 33
Store 2 56 21
This month(B)
dogs cats
Store 1 36 51
Store 2 48 37
Write a matrix for the average monthly sales by adding
1
A and B and then multiplying the result by .
2
1 ( A  B)  1  42 33  36 51 
2
2  56 21 48 37 

1  78 84 39 42

  

2 104 58 52 29
Store 1 sold an average of _39_ dogs and _42_ cats, while
Store 2 sold an average of _52_ dogs and _29_ cats.
Your Notes
Example 4
Solve a matrix equation
Solve the matrix equation for x and y.
 3x 1 1
 3   8  8 
4  

   

0
  0 6  2  2 y    8
Solution
Simplify the left side of the equation.
 3 x 1 1
4  

  0 6  2
 3    8  8 

 2 y    8
0 
 3x  1
 2    8  8 
4  
  

2 6  2 y    8
0

 8   8  8
12 x  4

 

8 24  8 y   8
0

Write original
equation.
Add matrices inside
parentheses.
Perform scalar
multiplication.
Equate corresponding elements and solve the two resulting equations.
_12x  4_  8
_24  8y_  0
x  _1_
y  _3_
Checkpoint Complete the following exercises.
3. Solve the matrix equation for x and y.
 3  2 x   1
 2  
  
1
7

  5y

 6    4 12 
  

4   8  6
x  6, y  1
Homework
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