Avon High School
Section: 9.4
ACE COLLEGE ALGEBRA II - NOTES
Multiplicative Inverses of Matrices and Matrix Equations
Mr. Record: Room ALC-129
Semester 2 - Day 17
The Multiplicative Identity Matrix
 1 3  1 0 
What happens if we multiply 

?
 4 6 0 1 
1
1 0 0  
0
1 0  
Matrices like 
, 0 1 0  , 

0 1  0 0 1  0

 
0
0 0 0
1 0 0 
are all called identity matrices.
0 1 0

0 0 1
The Multiplicative Inverse of a Matrix
Definition of the Multiplicative Inverse of a Square Matrix
Let A be an m n matrix. If there exists an n  n matrix and A1 (read: “A inverse”) such that
AA1  I n and A1 A  I n ,
then A1 is the multiplicative inverse of A.
Example 1
The Multiplicative Inverse of a Matrix
Show that B is the multiplicative inverse of A, where
 2 1
 1 1
A
and B  

.
1 1
 1 2 
Example 2
Finding the Multiplicative Inverse of a Matrix
Find the multiplicative inverse of
5 7 
A

 2 3
A Quick Method for Finding the Multiplicative Inverse of a 2 x 2 Matrix
Multiplicative Inverse of a 2 x 2 Matrix
a b 
1  d b 
, then A1 
.
If A  

ad  bc  c a 
c d 
The matrix A is invertible if and only if ad  bc  0 . If ad  bc  0, then A does not have a
multiplicative inverse.
Example 3
Using the Quick Method to Find the Multiplicative Inverse of a Matrix
Find the multiplicative inverse of
 3 2
A

 1 1 
Finding Multiplicative Inverses of a n x n Matrices with n Greater Than 2
Example 4
Finding the Multiplicative Inverse of a 3 x 3 Matrix
Find the multiplicative inverse of
 1 0 2
A   1 2 3 
 1 1 0 
Solving Systems of Equations Using Multiplicative Inverses of Matrices
Solving a System Using A-1
If AX  B has a unique solution, X  A1B. To solve a linear system of
equations, multiply A1 and B to find X.
Example 5
Using the Inverse of a Matrix to Solve a System
Solve the system by using A1 , the inverse of the coefficient matrix:
x
 2z  6
 x  2 y  3 z  5
x  y
6
Avon High School
Section: 9.4
ACE COLLEGE ALGEBRA II - NOTES
Multiplicative Inverses of Matrices and Matrix Equations
Mr. Record: Room ALC-129
Semester 2 - Day 18
Applications of Matrix Inverses to Coding
Encoding a Word or Message
1. Express the word or message numerically.
2. List the numbers in Step 1 by columns and form a square matrix. If you do
not have enough numbers to form a square matrix, put zeros in any
Encod
remaining spaces in the last column.
3. Select any square invertible matrix, called the coding matrix, the same
size as the matrix in Step 2. Multiply the coding matrix by the square
matrix that expresses the message numerically. The resulting matrix is the
Scalar
codedMultiplication
matrix.
4. Use the numbers, by columns, from the coded matrix in Step 3 to write the
encoded message.
Example 6
Encoding a Word
1 3 4 
a. Encode the message MATH IS COOL using the coding matrix A   2 1 3  . Use 0 for blank.


 4 2 1 
b. Decode the message 138, 81, 102, 101, 67, 109, 162, 124, 173, 210, 150, 165
1 3 4 
Given the coding matrix A   2 1 3  and a TI-Nspire calculator.


 4 2 1 