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Inversion
9.7 The Inverse of a Matrix
Objective: Students will solve systems using matrix
algebra, inverse matrices and identity matrices.
Matrix Inversion
1
1
B B  BB
Like a reciprocal
in scalar math
 I
Like the number one
in scalar math
Using matrix equations
Identity matrix: Square matrix with 1’s on the diagonal
and zeros everywhere else
 1 0


 0 1
1
0

0
0
1
0
2 x 2 identity matrix
0
0

1 
3 x 3 identity matrix
The identity matrix is to matrix multiplication as
1 is to regular multiplication!!!!
___
Multiply:
 1 0  5  2

 3 4 =
 0 1
5  2


3
4
 1 0

 =
 0 1
5  2


3
4
5  2


3
4
So, the identity matrix multiplied by any matrix
lets the “any” matrix keep its identity!
Mathematically,
IA = A and AI = A !!
Using matrix equations
Inverse Matrix: 2 x 2
A 
 a b


 c d
A
1

1  d  b
ad  bc  c a 
In words:
•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.
Using matrix equations
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
Find the inverse matrix.
 8  3


 5
2
Inverse =
Matrix A


1  Matrix 
det  Reloaded


Det A = 8(2) – (-5)(-3) = 16 – 15 = 1
1
=
1
2 3


5 8
=
2 3


5 8
What happens when you multiply a matrix by its inverse?
1st:
What happens when you multiply a number by its inverse?
A & B are inverses. Multiply them.
 8  3 2 3

 

 5
2 5 8
=
So, AA-1 = I
 1 0


0 1
1
7
7
Why do we need to know all this? To Solve Problems!
Solve for Matrix X.
 8  3


 5
2
X
 4  1


 3
1
=
We need to “undo” the coefficient matrix.
2 3  8  3

 

5 8  5
2
 1 0


0 1
X
X
X
=
=
Multiply it by its INVERSE!
2 3  4  1

 

5 8  3
1
  1 1


 4 3
1
= 
1


 4 3
Using matrix equations
You can take a system of equations and write it with
matrices!!!
3x + 2y = 11
2x + y = 8
becomes
3 2


2 1
x 
11
y
 
  = 8
Coefficient Variable
matrix
matrix
Answer matrix
Using matrix equations
Example: Solve
3 2


2 1
x 
11
y =  8 
 
 
for x and y .
Let A be the coefficient matrix.
Multiply both sides of the equation by the inverse of A.
A
1

3

2
2 1
1
1  1  2
=
 1  2 3 
2
 1


 2  3
2
 1
= 

 2  3
2
3 2 x   1

 y = 

2 1    2  3
 1 0  x   5 

  y  =   2
0 1    
x   5 
 y  =   2
   
11
8
 
Using matrix equations
Wow!!!!
x = 5; y = -2
It works!!!!
Check:
3x + 2y = 11
3(5) + 2(-2) = 11
2x + y = 8
2(5) + (-2) = 8
You Try…
Solve:
4x + 6y = 14
2x – 5y = -9
(1/2, 2)