Download Matrix Methods

Math 2/Unit 2/Lesson 2/ TOOLKIT: Matrix Methods (Investigation 1)
In this investigation, you learned how to multiply a row matrix by another matrix and how to interpret
the product.
[1 x 4] x [4 x 2] = [1 x 2]
3
7
[3 6 -5 9] 
3

6
5
8 
= [90 171]
- 9

7
To multiply matrices the
number of columns in the
first matrix must be the
same as the number of rows
in the second matrix
3(3) + 6(7) + (-5)(3) + 9(6) = 90
3(5) + 6(8) + (-5)(-9) + 9(7) = 171
The result is the number of
rows of the first matrix and
the number of columns in
the second matrix
Math 2/Unit 2/Lesson 2/ TOOLKIT: Matrix Methods (Investigation 2)
Matrix multiplication can be useful but only in certain situations.
[2 x 3] x [3 x 2] = [2 x 2]
1 2 
1 2 3 
 22 28
4 5 6  3 4   49 64


 5 6  


1(1) + 2(3) + 3(5) = 22
1(2) + 2(4) + 3(6) = 28
4(1) + 5(3) + 6(5) = 22
4(2) + 5(4) + 6(6) = 28
Math 2/Unit 2/Lesson 2/ TOOLKIT: Matrix Methods (Investigation 3)
In this investigation, you explored how powers of an adjacency matrix for a digraph and sums of the powers
could be used to analyze the digraph and the situation it models.
Squaring a Matrix
2
Multiplying a matrix by
itself
1 2
1 2 1 2 7 10 
3 4   3 4  3 4   12 22




 

1(1) + 2(3) = 7
1(2) + 2(4) = 10
3(1) + 4(3) = 15
3(2) + 4(4) = 22
Math 2/Unit 2/Lesson 3/ TOOLKIT: Matrix Methods (Investigation 1)
In this investigation, you examined properties of matrices and their operations and compared them with
corresponding properties of real numbers.
Adding/Subtracting
Matrices
When you add or subtract matrices, the sizes must be exactly the same.
[2 x 2] + [2 x 2] = [2 x 2]
1 2 5 6 6 8 
3 4   7 8   10 12

 
 

Adding Matrices is
Commutative
1 2 5 6 5 6 1 2
3 4   7 8   7 8   3 4 

 
 
 

Additive Identity or Zero
Matrix is a matrix with all
zeros
1 2 0
3 4   0

 
Additive Inverse Matrix is
the matrix that adds to the
original matrix to create a
zero matrix
1 2  1 - 2 0
3 4   - 3 - 4   0

 
 
Multiplicative Identity is the
1 0
matrix. 

0 1 
This matrix has all ones on
the main diagonal and zeros
everywhere else. It also
must be a square matrix and
it is commutative.
0 1 2

0 3 4 
1 2 1 0 1 2
3 4  0 1  3 4 


 

1 0 1 2 1 2
0 1 3 4   3 4 


 

0
0
Inverse Matrix of A, where
d - b 
A= 
 is
- c a 
1 d - b 
A 1 
ad  bc - c a 
5 3 
If A= 
 , then
 4 4
4 - 3 
1
A 1 
5(4)  3(4) - 4 5
1 4 - 3 
A 1 
20  16 - 4 5
1 4 - 3 
4 - 4 5
 -3 
1 4 
1
A 

- 1 5 
4 

A 1 
To find A-1 with your calculator
use the x-1 button
If you multiply [A] [A-1] you
should get the identity matrix
1 0
0 1 


 -3 
1 4 
5 3 
1
If A= 
 then
 and A  
 4 4
- 1 5 

4 
 -3 
1
5
3

 
4  = 1 0
 
 4 4 


  - 1 5  0 1 
4 
