Download 1.3p Determinants, Inverses

Warm Up
Solve for x and y:
 3 2x   1 6   4 12 
2  





 1 7   5y 4    8 6
x6
y  1
What would happen to this problem if there
weren’t parentheses?
**Watch your signs**
Determinants
A real number associated with
any square matrix A, denoted by
detA
or
|A|
Determinant of a 2x2
a bO a b
L
det M P 
cb
=
ad
c dQ c d
N
Ex. Evaluate the determinant:
1 4O
L
M
P
2 7Q
N
1 4
2 7
=1(7) — 2(4)
=7-8
= -1
Ex. Evaluate the determinant:
 7  2
2 3 


7 2
2 3
=7(3) - 2(-2)
= 21 + 4
= 25
What is an ADDITIVE…..
IDENTITY:
What can I add to a matrix by to keep it the same?
 2 7  ? ?  2 7 
 1 3  ? ?   1 3
 


 
? ?  0 0 
? ?   0 0 

 

What is an ADDITIVE…..
INVERSE:
What can I add to a matrix to eliminate all elements?
 2 7  ? ?  0 0 
 1 3  ? ?  0 0 
 


 
? ?  2 7 
? ?   1 3 

 

What is a MULTIPLICATIVE…..
IDENTITY:
What can I multiply a matrix by to keep it the same?
2 7 
 1 3


? ?
2 7 
? ?   1 3




? ?  1 0 
? ?   0 1

 

***We call this the IDENTITY matrix***
What is a MULTIPLICATIVE…..
INVERSE:
What can I multiply a matrix by to get the Identity Matrix?
2 7 
 1 3


? ?  1 0 
? ?   0 1


 
? ?  2 7 
? ?   1 3 

 

1
*** Multiplicative Inverse = A
-1
***
How do I actually find
the Inverse Matrix?
• You need to find the determinant!!!
The Inverse of a 2x2 Matrix
a b 
A

 c d
-1
A =
1
A
 d  b
 c a 


***If ad-cb=0,
then the
matrix has no
inverse!!!!
 d  b
1



ad  bc  c a 
As long as ad-cb =0
Ex. Find A-1, if it exists.
2 3
A 

5 7 
A-1=
det A  14 15  1
1  7 3 


1  5 2 
 7 3 
 1


5
2


 7 3 
A 

 5 2 
1
Ex. Find A-1, if it exists.
 2 1
A 

4 0
A-1=
det A  0 4  4
1  0 1


4  4 2 
1 

0


4
1
A 

1  1

2 
Ex. Find A-1, if it exists.
3 4 


A  2 6


1 0 
det A 
NOT POSSIBLE
A-1= Does not exist,
because it’s not a square
matrix!
Homework:
***Worksheet***
1.3