Download Determine the Inverse of a Matrix

College Algebra
Chapter 6
Matrices and Determinants
and Applications
Section 6.4
Inverse Matrices and
Matrix Equations
Concepts
1. Identify Identity and Inverse Matrices
2. Determine the Inverse of a Matrix
3. Solve Systems of Linear Equations Using the Inverse
of a Matrix
Identify Identity and Inverse Matrices
The identity matrix In is the n  n square matrix with 1’s
along the main diagonal and 0’s for all other elements.
1 0 
Identity matrix of order 2 = I 2  

0
1


1 0 0 


Identity matrix of order 3 = I 3  0 1 0


0 0 1 
For an n  n square matrix A: AI n  A and I n A  A
(Identity property of matrix multiplication)
Identify Identity and Inverse Matrices
Let A be an n  n matrix and In be the identity matrix of
order n. If there exists an n  n matrix A–1 such that
AA1 = I n and A1 A  I n
then A–1 is the multiplicative inverse of A.
Example 1:
  14
 5 2 
Determine if A  
and B   1

 1 2
 8
are inverses.
 14 
 85 
Example 1 continued:
Concepts
1. Identify Identity and Inverse Matrices
2. Determine the Inverse of a Matrix
3. Solve Systems of Linear Equations Using the Inverse
of a Matrix
Determine the Inverse of a Matrix
Let A be an n  n matrix for which A–1 exists,
and let In be the n  n identity matrix.
To find A–1:
Step 1:
Write a matrix of the form  A I n  .
Step 2:
Perform row operations to write the matrix in
the form  I n B  .
Step 3:
The matrix B is A–1.
Determine the Inverse of a Matrix
Note: Not all matrices have a multiplicative inverse.
If a matrix A is reducible to a row-equivalent matrix
with one or more rows of zeros, the matrix does not have
an inverse, and we say that the matrix is singular.
A matrix that does have a multiplicative inverse is said
to be invertible or nonsingular.
Example 2:
 1 0 1
Given A   0 1 1  , find A-1 if possible.


 2 1 2 
Example 2 continued:
Example 3:
3
 2 5
Given A   4 10 6 , find A-1 if possible.


 0
1 12 
Determine the Inverse of a Matrix
Formula for the inverse of a 2  2 invertible matrix:
a b 
Let A  
be an invertible matrix.

c d 
Then the inverse A1 is given by:
 d b 
1
A 


ad  bc  c a 
1
Example 4:
 29 3
-1.
Given A  
,
find
A

19
2


Concepts
1. Identify Identity and Inverse Matrices
2. Determine the Inverse of a Matrix
3. Solve Systems of Linear Equations Using the Inverse
of a Matrix
Solve Systems of Linear Equations Using the Inverse of
a Matrix
A system of linear equations written in standard form
can be represented by using matrix multiplication.
For example:
 x  2 y – z  4
corresponding

 5 x  6 y – 3 z  2 matrix equation
12 x – 2 y  2 z  1

A ∙ X = B




  x 
  y  
  
  z  




Solve Systems of Linear Equations Using the Inverse of
a Matrix
A ∙ X = B




coefficient
matrix
  x 
  y  
  
  z  
column matrix of
variables




column matrix of
constants
Solve Systems of Linear Equations Using the Inverse of
a Matrix
AX = B
To solve this equation,
the goal is to isolate X.
A–1AX = A–1B
Multiply both sides by A–1
(provided that A–1 exists).
X = A–1B
Example 5:
Solve the system by using the inverse of the coefficient
matrix.
3x  5 y  91
2 x  6 y  14
Example 5 continued: