Download day 3 - Cobb Learning

Identity & Inverse Matrices
Identity
What does “identity” mean to you?
What is the multiplicative identity
for the real numbers?
In other words, 5 * __= 5?
The identity for multiplication is 1
because anything multiplied by 1 will
be itself.
Inverses
What does “inverse” mean to you?
What is the inverse of multiplication?
What do we multiply by to get the
identity?
In other words, 5 * ___=1?
a *
-1
a =
1
Any number multiplied by its
inverse will be the identity.
Identity Matrix
The multiplicative identity for
matrices is a square matrix with
ones on the main diagonal and
zeros everywhere else.
1 0
I 

0 1 
1 0 0


I  0 1 0 
0 0 1
Identity Matrix
Just like 5*1 = 5…
AI= A
2
1
 8
IA= A
 3 1 0  2






17  0 1  1
 
  8
 3

17 

Or
1 0  2  3  2  3




0 1 1 17  1 17 
  8


  8
Identity Matrix
Any matrix multiplied by its inverse
will be the identity matrix.
A *
-1
A
-1
A =
I
*A = I
2x2 Identity Matrix
1 0
I 

0 1 
3x3 Identity Matrix
1 0 0


I  0 1 0 
0 0 1
Ex. 1 Determine whether A and B are
inverses.
 2  1
 2 3
A
2
B


1
3
6




3
YES
Ex. 2 Determine whether A and B are
inverses.
 4 3
 5  3
B
A


 7 5
 7 4 
NO
The Inverse of a 2x2 Matrix
a b 
A

c d 
-1
A =
1
A
If ad-cd=0, then the matrix
has no inverse!!!!
 d b 
 c a 


1  d b 



ad  bc  c a 
As long as ad-cb =0
Ex. 3 Find A-1, if it exists.
 2 3
A

5 7 
-1
A =
1  7  3


14  15  5 2 
 7 3 
A 

 5  2
1
Ex. 4 Find A-1, if it exists.
2 1
A

 4 0
-1
A =
1 

0
0

1

 
1 
4




1
 4  4 2  1  
2

Ex. 5 Find A-1, if it exists.
3 4 


A   2  6
1 0 
Does not exist,
because it’s not
square.
Now let’s learn how to
use our calculator!!!
Find the inverse!
 2 3
A

5 7 
2 1
A

 4 0
Yes, now you can add, subtract, multiply,
and find the determinant in you calculator!!

Solving Systems using
Matrices and Inverses
Solving Matrix Equations
Suppose ax = b
How do you solve for x?
We cannot divide
matrices, but we
can multiply by the
inverse.
A-1 AX =A-1 B
IX = A-1B
X = A-1B
Solving a Matrix Equation
Solve the matrix equation AX=B for
the 2x2 matrix X
 4  1
 8  5
X

 3 1 
 6 3 




X = A-1B
 2  2
X 

 0  3
Ex. Solve
 3 4 
3 8 
X

 5  7
 2  2




 29  48
X 

  21  34
Solving Systems Using
Inverse Matrices
5x  2 y  3
4x  2 y  4
Setting Up the Matrices
• Matrix A will be the coefficients of
the system
• Matrix X will be the variables
• Matrix B will be constants (what
the system of equations are equal
to)
Matrix Equation
A linear system can be written as a
matrix equation AX=B
5  4  x  8
1 2   y   6

   
Constant
Coefficient
matrix
Variable
matrix
matrix
5x  4 y  8
1x  2 y  6
Example 1
5x  4 y  8
1x  2 y  6
5  4  x  8
1 2   y   6

   
Example 2:
Use matrices to solve the linear system
5x  2 y  3
4x  2 y  4
5 2  x 
4 2  y  

  
3
4
 
Type in [A]-1 [B]
Find the inverse
 1 1
3
 x 



5  
 y  
  2
 4

2
(-1, 4)
Example 3:
Use matrices to solve the linear system
4x  2 y  8
x  2 y  12
 4 2  x   8 
1 2   y   12

   
Type in [A]-1 [B]
Find the inverse
 x   .2 .2  8 
 y    .1 .4  12
  
 
(4, 4)
Example 4:
Use matrices to solve the linear system
x  y  2z  3
2 x  y  3z  4
4 x  3 y  z  18
2  x  3 
1 1
2  1 3   y     4 

  

4  3  1  z   18
Type in [A]-1 [B]
(-2, 3, 1)
Example 5:
Use matrices to solve the linear system
2x  z  2
5x  y  z  5
x  2 y  2z  0
 2 0 1   x   2
 5 1 1   y   5 

   
 1 2 2  z  0 
Type in [A]-1 [B]
(2, 3, -2)
Let’s apply this…
You have $18 to spend for lunch during a 5
day school week. It costs you $1.50 to make
lunch at home and $5 to buy lunch. How
many times each week do you make a lunch
at home?
x y 5
1.5 x  5 y  18
(2, 3)
 1 1  x   5 
1.5 5  y   18

   
Type in [A]-1 [B]
You make lunch at home 2 times a week.
A word problem…!!
• A small corporation borrowed $1,500,000 to
expand its product line. Some of the money was
borrowed at 8%, some at 9% and some at 12%.
How much was borrowed at each rate if the
annual interest was $133,000 and the amount
borrowed at 8% was 4 times the amount
borrowed at 12%?
$800,000 at 8%
$500,000 at 9%
$200,000 at 12%
Homework