Section 8.4
Multiplicative Inverses of Matices
and Matrix Equations
The Multiplicative Identity
Matrix
1 0
The Multiplicative Identity matrix is I= 
 for 2 × 2 matrices.
0 1
That means that AI=A and IA=A
The Multiplicative Inverse
of a Matrix
If a square matrix has a multiplicative inverse, it is
said to be invertible or nonsingular. If a square matrix
has a multiplicative inverse, the inverse is unique. If
a square matrix has no multiplicative inverse, it is
called singular.
Example
2 1
Find the multiplicative inverse of A= 

7
4


A Quick Method for Finding the
Multiplicative Inverse of a 2 x 2
Matrix
Example
Find the multiplicative inverse of A using the Quick
Method to find the inverse.
 −3 2 
A= 

1
−
4


Example
Find the multiplicative inverse of A using the
Quick Method. Check your work using your
 −2 3 
calculator. A= 

 1 5
Example
Show that A does not have an inverse. First by calculations,
then use your calculator and see what you get for an answer.
 −3 2 
A= 

6
−
4


Finding Multiplicative Inverses
of n x n Matrices with n Greater
Than 2
If ad-bc=0 then the matrix has no multiplicative
inverse.
Example
Find the inverse function without a calculator for
 −2 1 
−1
−1
A= 
.
Show
that
A
A
=
I
and
A
A=I 2 .

2
 −1 3 
Find the multiplicative inverse of A by row calculations,
then check your work using the calculator.
 1 1 0


A=  −1 3 4  .
 0 4 3


Example
Find the multiplicative inverse matrix of A using row
calculations. Then check your answer using your calculator.
 1 2 −2 


A=  0 1 −1  .
 2 −1 0 


Solving Systems of Equations
Using Multiplicative Inverses
of Matrices
Solve the system using A −1 , the inverse of the coefficient matrix.
x+z=3
x-y=-2
x-y+2z=2
Example
Solve the system by using A −1 , the inverse of the
coefficient matrix.
x+ y- z =2
2y+ z=3
−x − 2 y = 1
Applications of Matrix Inverses
to Coding
A cryptogram is a message written so that no one other than the intended recipient can understand it. To encode a message, we begin by assigning a number
to each letter in the alphabet: A=1, B=2, C=3, . . .Z=26, and a space =0. The
numerical equivalent of the word ATTITUDE=1,20,20,9,20,21,4,5
The numerical equivalent of the word MATH is 13,1,20,8. The numerical
equivalent of the message is then converted into a matrix. Finally, an
invertible matrix can be used to convert the message into code. The
multiplicative inverse of this matrix can be used to decode the message.
Encoding the Word MATH
Decoding a Word
Example
 4 −1
For the word CASH which is 3,1,19,8, use the coding matrix A= 

−
3
1


1 1 
to encode the word. Then use the matrix A −1 = 
 to decode the given
3 4
word. The problem has already been started for you.
 4 −1  3 19 


=
−
3
1
1
8



 1 3
Find the multiplicative inverse of A= 
.
 −2 2 
 2 −3 
(a) 

2
1


1 0
(b)  0 1 


 1 −3 


(c)  4 8 
1 1 


4 8 
 1 3 
−

(d)  2 4 
− 1 − 1 


 2
4
 1 3 −1 
Find the multiplicative inverse of A= 
.
 −2 2 1 
(a)  2 −3 1 


2
1
−
1


1 0 2
(b) 

0
1
1


 1 4 −1
(c)  −1 2 0 


(d) No inverse exists