Download Finding the Inverse of a Matrix

Finding the Inverse of a
Matrix
Properties of Matrices
We have discovered that the commutative
property for multiplication does not work for
matrix multiplication. Let’s consider some
of the other properties of real numbers. Is
there a multiplicative identity for matrices?
Is there a multiplicative inverse for
matrices?
The Multiplicative Identity
The multiplicative identity for real numbers
is the number 1. The property is:
If a is a real number, then a x 1 = 1 x a = a.
In terms of matrices we need a matrix
that can be multiplied by a matrix (A) and
give a product which is the same matrix
(A).
The Multiplicative Identity
This matrix exists and it is called the
identity matrix. It is named I and it comes
in different sizes. It is a square matrix with
all 1’s on the main diagonal and all other
entries are 0.
1 0 0
1 0
0 1 0
I

I2  
3



0 1 
0 0 1
The Multiplicative Identity
Multiply AI
 2 5
A 

4
0


 2 5 1 0
 4 0 0 1



a11= (-2)(1) + (5)(0) = -2
a12= (-2)(0) + (5)(1) = 5
a21= (4)(1) + (0)(0) = 4
a22= (4)(0) + (0)(1) = 0
 2 5


 4 0
The Identity Matrix for Multiplication
Let A be a square matrix with n rows
and n columns. Let I be a matrix with
the same dimensions and with 1’s on
the main diagonal and 0’s elsewhere.
Then AI = IA = A
The Multiplicative Identity
Give the multiplicative identity for matrix B.
1 0 0 0 
0 7 4 9 
 0 1 0 0
 3 7  9 2


I 
B
 0 0 1 0
0 1  4 7 




0 0 0 1 
6 0 4 1 
This identity matrix is I4.
The Multiplicative Inverse
For every nonzero real number a, there is a
real number 1/a such that a(1/a) = 1.
In terms of matrices, the product of a
square matrix and its inverse is I.
 3 1  1  1 3(1)  1( 2) 3( 1)  1(3)  1 0
2 1  2 3   2(1)  1( 2) 2( 1)  1(3)  0 1


 
 

The Inverse of a Matrix
Let A be a square matrix with n rows
and n columns. If there is an n x n
matrix B such that AB = I and BA = I,
then A and B are inverses of one
another. The inverse of matrix A is
denoted by A-1.
The Inverse of a Matrix
To show that matrices are inverses of one
another, show that the multiplication of the
matrices is commutative and results in the
identity matrix.
Show that A and B are inverses.
2 3
 5  3
A 
and B  


3 5
 3 2 
The Inverse of a Matrix
2 3  5  3
AB  



 3 5   3 2 
2(5)  3( 3) 2( 3)  3( 2)


 3(5)  5(3) 3( 3)  5( 2) 
1 0 


and 
0
1


The Inverse of a Matrix
 5  3 2
BA  
 3
2
3



5( 2)  ( 3)( 3)

  3( 2)  2(3)
1 0 


0 1 
3
5
5(3)  ( 3)( 5)
 3(3)  2(5) 
Finding the Inverse of a Matrix Method 1
Use the equation AB = I.
1 2
a b 
Let A  
and B  


3
5
c
d




Write and solve the equation:
1 2 a b  1 0
3 5  c d   0 1


 

Inverses – Method 1, cont.
1 2 a b  1 0
3 5  c d   0 1


 

 a  2c b  2d  1 0
3a  5c 3b  5d   0 1

 

a  2 c  1

3a  5c  0
b  2d  0

3b  5d  1
a  5 and c  3
b  2 and d  1
Inverses – Method 1, cont.
 5 2 
So the inverse of A = 

 3  1
We can check this by multiplying A x A-1
1 2  5 2  1( 5)  2(3) 1(2)  2( 1) 
3 5  3  1  3( 5)  5(3) 3(2)  5( 1)


 

1 0


0
1


Finding the Inverse with a Calculator
To find the inverse of a matrix using the
calculator, enter the matrix into the
calculator and use the x-1 key.
Finding the Inverse with a Calculator
Find the inverse of each matrix using the
calculator.
 2  1 1
B    1 3 4


 2 1 0
8 4
C

6 3
Finding the Inverse with a Calculator
This error message
means that the matrix
does not have an
inverse.
A matrix that does not have an inverse is
called an invertible matrix.
Determinants
Each square matrix can be assigned
a real number called the determinant
of the matrix. It is denoted by the
symbol
.
a b 
If A  

c d 
a b
c d
means the
determinant
of A.
Determinants
The determinant of a 2 x 2 matrix is
found as follows:
a b
 ad  cb
c d
Determinants
Find the determinant
of the matrix.
7 8 
G

6 7
7 8
 7(7)  6(8)  49  48  1
6 7
Determinants
Find the determinant of the matrix.
1 1 
H 

2
2


1 1
 1(2)  2(1)  0
2 2
If the determinant
of a matrix = 0, the
matrix does not
have an inverse.
Matrix H is
invertible.
Determinants can be used to find the
inverse of a matrix.
a b 
If A  
and
det
(
A
)

0
,
then

c d 
1  d  b
1
A 


det ( A)  c a 
Determinants can be used to find the
inverse of a matrix.
 d  b
 c a  is called the adjoint of the

 original matrix. Notice it is
found by switching the entries on the
main diagonal and changing the signs of
the entries on the other diagonal.
Find the multiplicative inverse of:
1 2
A 

3
4


1 2
 1(4)  3(2)  2
3 4

2
1


4

2


1
1
A 
 3




1
 2  3 1   2
2
We can check to see if we are correct by
multiplying. Remember that AA-1 = I
1 
1 2  2
3 4   3
1 

 2
2
1( 2)  2(3 / 2) 1(1)  2( 1 / 2) 


3
(

2
)

4
(
3
/
2
)
3
(
1
)

4
(

1
/
2
)


1 0


0
1


Find the inverse using determinants.
1 3
1 1


2 1
0 3


3 
 1
2
 2
 1 2  1 2 
1 
1
6
 2
1 
 0
3
Find the inverse
 4 8 
 2  4


No inverse
Recall that when the determinant
of a matrix is 0 the matrix will not
have an inverse because division
by 0 is undefined.
Finding the determinant of a 3 x 3
matrix
Finding the determinant of a 3x3
matrix.
One way to find the determinant of a 3x3
matrix is the formula below.
a b
d e
g h
c
e
f a
h
i
f
d
b
i
g
f
d e
c
i
g h
Find the determinant using the formula
2 0 5
3 1 5
0 2 4
Find the determinant using the formula
2
0
5
1 5
3 5
3 1
3 1 5  2
0
5
2 4
0
4
0 2
0 2 4
 21( 4)  2( 5)  0 3( 4)  0( 5)  5 3( 2)  0(1)
 2(14)  0( 12)  5( 6)
 28  30
 2
Find the determinant using the formula
 2 1 1
1 2 3
4
1 2
Find the determinant using the formula
 2 1 1
2 3
1 3
1 2
1  2 3  2
 ( 1)
1
1 2
4 2
4 1
4
1 2
 2 2(2)  1(3)  11(2)  4(3)  11(1)  4( 2)
 2( 7)  1( 10)  1(9)
 14  10  9
 13