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Inverse Matrices (2 x 2)
How to find the inverse of a 2x2
matrix
Inverse of a number
When we are talking about our natural
numbers, the inverse of a number is it’s
reciprocal. When we multiply a number
by it’s inverse we get 1.
For example:
3 1  1
3
4  0.25  1
k  k 1  1
Inverse of a matrix
What do you think we would get if we
multiplied a matrix by it’s inverse? Try
it on your calculator.
1
A A  I
A matrix multiplied by its inverse always
gives us an identity matrix.
Inverse of a matrix
Not all matrices have an inverse.
If the determinant of a matrix is 0,
then it has no inverse and is said to be
SINGULAR.
All others are said to be NON-SINGULAR
Inverse of a matrix
Which of these have an inverse?
3 2
3 2
1 2
3 4
Finding Inverses 2x2
 8 10 
A

 3 4 
1
A A  I
a b 
Let A-1 = 

c
d


So
AA-1
=I
 8  10 a b  1 0







3
4
c
d
0
1


 

Multiplying out gives..
 8a  10c 8b  10d  1 0





3
a

4
c

3
b

4
d
0
1

 

8a  10c  1
 3a  4c  0
8b  10d  0
 3b  4d  1
Can you solve these to work out A-1?
 2 5
A 

1.5 4
1
Finding Inverses 2x2
There is an alternative method.
a b 
A

c
d


1  d b 
A 
ad  bc  c a 
1
In words:
ad-bc represents det(A). What
would happen if this was zero?
1
det( A)
d
a
-b
b
-c
c
d
a
•Take the original matrix.
•Switch a and d.
•Change the signs of b and c.
•Multiply the new matrix by 1 over the determinant of the original matrix.
Finding Inverses 2x2
Example: Find the inverse of A.
A 
A
A
1
1
4
 2


 4  10 

 10  4 
1
2 
(2)(10)  ( 4)(4)  4

1  10  4 
2  =
 4  4
5

1
2


1
 1  

2
Finding Inverses 2x2