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What is AI?
 2  1 2
0 1 5   A


2  2 3
1 0 0


I 3  20 1 12 0
What is IA? 0 1 5  A



0
0
1


2

2
3
 1 2



2
A  0 1 5
2  2 3
Multiplying a
matrix by the
identity gives the
matrix back again.
1 0 0


I 3  0 1 0
0 0 1
an n  n matrix with ones on the main diagonal
and zeros elsewhere
Can we find a matrix to multiply the first matrix by to
get the identity?
1

 3  1  1  2  1
 ? 3   
4

22
0




2 
1


1


3

1
1





2





3 4
2
0


2

2 

0

1
0

1
Let A be an n n matrix. If there exists a matrix B
such that AB = BA = I then we call this matrix the
inverse of A and denote it A-1.
a b 
c d 


Finding the Inverse of a 2x2 matrix
Step-1 First find what is called the Determinant
This is calculated as ad-bc
Step-2 Then swap the elements in the leading diagonal
d b
c a 


d  b 
 c a 


Step-3 Then negate the other elements
Step-4 Then multiply the Matrix by 1/determinant
1 d  b 
ad  cb  c a 
inverse matrix
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Example Find Inverse of A
Step 1 – Calc Determinant
 4 8
Determinant (ad-cb) = 4x3-8x1 = 4

A  
 1 3
Step 2 – Swap Elements on leading diagonal
step2
3 8


1
4


Step 3 – negate the other elements
step3
 3 8


4
 1
Step 4 – multiply by 1/determinant
step4
1  3 8


4
4  1
check
 0.75  2
1
A 


0
.
25
1


AA
inverse matrix
1
4 8  0.75  2

 

1 3   0.25 1 
3  2
 
0.75  0.75
 8  8
1 0
  0 1
23
 
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Find the inverses and check them
 2 6

A  
1 5
1  5  6   1.25  1.5 
  

A  
4   1 2    0.25 0.5 
  5 20 

B  
 1 2 
1  2  20   0.2  2 
  

B  
10  1  5   0.1  0.5 
2 2 

C  
 0  1
1
1
1   1  2   0.5 1 

  

C 
2   0  1
2 0
1
inverse matrix
6
We can use A-1 to solve a system of equations
x  3y  1
2x  5 y  3
To see how, we can re-write a
system of equations as matrices.
Ax  b
coefficient
matrix
variable
matrix
1 3
2 5


 x
 y
 
constant
matrix
1
  
3
Why will it help us solve equations?
Because if we can express a system of equations in the
form
Ax  b
Then we can multiply both sides by the inverse matrix
1
1
A Ax  A b
And we can then know the values of X because
1
A A I
x A b
1
inverse matrix
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Solving systems of equations
We can use our 2x2 matrices to express these systems
of equations
x y
x  3y
 240 

0 
Becomes in matrix
form
1 1  x1   240 

   

1  3  x 2   0 
constants
from the
left hand
side
UNKNOWNS
X ~ x1
Y ~ x2
inverse matrix
constants
from the
right hand
side
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Your calculator can compute inverses and
determinants of matrices.
For the inverse:
Step 1: enter the matrix
into the calculator
Step 2: Pull up matrix
entered onto the screen
Step 3: hit the x^-1 button
and enter (this will give
you the inverse matrix).
Now using the previous problem we will allow the
calculator to do the work.
Enter this as
matrix A in
the calc
1 1  x1   240 

   

1  3  x 2   0 
Enter this as
Matrix B in the
calc
Your Turn solve the following
3x +4y = 5
5x = 7-6y
x+7y = 1.24
3y -x = 0.76
8x = 3y -1
x+y =-7
 3 4  x   5 

    
 5 6  y   7 
 1 7  x   1.24 

   

  1 3  y   0.76 
 8  3  x    1 

    
 1 1  y    7 
inverse matrix
Answer x = -1 y = 2
Answer x = -0.16 y = 0.2
Answer x = -2 y = -5
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