Download Chapter 2 Systems of Linear Equations and Matrices

Chapter 2
Systems of Linear Equations
and Matrices
Section 2.5
Matrix Inverses
What is a Matrix Inverse?


The inverse of a matrix is comparable
to the reciprocal of a real number.
The product of a matrix and its
identity matrix is always the matrix
itself.
In other words, multiplying a matrix
by its identity matrix is like multiplying
a number by 1.
Multiplicative Identity


The real number 1 is the multiplicative
identity for real numbers:
for any real number a, we have
a•1 = 1•a = a
In this section, we define a multiplicative
identity matrix I that has properties
similar to those of the number 1.
We use the definition of this matrix I to
find the multiplicative inverse of any
square matrix that has an inverse.
Identity Matrix


If I is to be the identity matrix, both
of the products AI and IA must
equal A.
The identity matrix only exists for
square matrices.
Examples of Identity Matrices
Determining if Matrices are
Inverses of Each Other



Recall that a number multiplied by its
multiplicative inverse yields a product of 1.
Similarly, the product of matrix A and its
multiplicative inverse matrix A 1 (read “Ainverse”) is I, the identity matrix.
So, to prove that two matrices are inverses
of each other, show that their product,
regardless of the order they’re multiplied, is
always the identity matrix.
Example 1

Prove or disprove that the matrices below are
inverses of each other.
a.)
b.)
5 7
 3 7 

 and 

 2 3
 2 5 
 1 2 
 5 2 

 and 

 3 5 
 3 1 
0
c.)  0

1

1
0
1
0 
1


2  and  1
0
0 


0
0
1
1

0
0

Finding the Inverse of a Matrix
Row Operations on Matrices
Example 2

Find the inverse, if it exists, for each
matrix.
a.)
c.)
 1 2 


 2 1
 5 10 


 3 6 
b.)
 1 2 


3 4
Shortcut for Finding the Inverse of
a 2 x 2 Matrix
If a matrix is of the form
a b 


c d
then the inverse can be found by
calculating:
1  d b 


ad  bc  c a 
Note: ad – bc ≠ 0.
Example 3

Find the inverse of the matrix below using
the shortcut method.
 4 2


5 3
Solution to Example 3
To find the inverse
of the matrix
use the formula
and simplify.
 4 2


5 3
1  d b 


ad  bc  c a 
 3 2 
1


4(3)  2(5)  5 4 
Solution to Example 3 (continued)
 3
1

4(3)  2(5)  5
1 3

2  5
 3
 2

 5

 2
2 

4 

1
1
 A

2 

2 

4 