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Chapter 9Matrices and Determinants
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1
9.4 Multiplicative Inverses of Matrices and Matrix Equations
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2
• Find the multiplicative inverse of a square matrix.• Use inverses to solve matrix equations.• Encode and decode messages.
Objectives:
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The Multiplicative Identity Matrix
The square matrix with 1’s down the main diagonal from upper left to lower right and 0’s elsewhere is called the multiplicative identity matrix of order n. This matrix is designated by In.
For example,
n n
2
1 0,
0 1I
3
1 0 0
0 1 0 .
0 0 1
I
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Definition of the Multiplicative Inverse of a Matrix
If a square matrix has a multiplicative inverse, it is said to beinvertible or nonsingular. If a square matrix has no multiplicative inverse, it is called singular.
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Example: The Multiplicative Inverse of a Matrix
Show that B is the multiplicative inverse of A, where
2 1
1 1A
1 1
1 2B
2 1 1 1
1 1 1 2AB
2(1) 1( 1) 2( 1) 1(2)
1(1) 1( 1) 1( 1) 1(2)
1 0
0 1
1 1 2 1
1 2 1 1BA
1(2) 1(1) 1(1) 1(1)
1(2) 2(1) 1(1) 2(1)
1 0
0 1
AB = BA = I. Thus, B is the multiplicative inverse of A.
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A Quick Method for Finding the Multiplicative Inverse of a 2 x 2 Matrix
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Example: Using the Quick Method to Find a Multiplicative Inverse
Find the multiplicative inverse of 3 2
1 1A
3 2
1 1A
a b
dc
1 1 d bA
ad bc c a
1 1 ( 2)13(1) ( 2)( 1) ( 1) 3
A
1 211 1 3
1 2
1 3
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Procedure for Finding the Multiplicative Inverse of an Invertible Matrix
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Example: Finding the Multiplicative Inverse of a 3 x 3 Matrix
Find the multiplicative inverse of
Step 1 Form the augmented matrix
1 0 2
1 2 3
1 1 0
A
3 .A I
1 0 2 1 0 0
1 2 3 0 1 0
1 1 0 0 0 1
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Example: Finding the Multiplicative Inverse of a 3 x 3 Matrix (continued)
Find the multiplicative inverse of
Step 2 Perform row operations on to obtain a matrix of the form
1 0 2
1 2 3
1 1 0
A
3A I
3 .I B1 0 2 1 0 0
1 2 3 0 1 0
1 1 0 0 0 1
replace row 2 by 1 2R R
1 0 2 1 0 0
0 2 5 1 1 0
1 1 0 0 0 1
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Example: Finding the Multiplicative Inverse of a 3 x 3 Matrix (continued)
Step 2 (cont)
1 0 2 1 0 0
0 2 5 1 1 0
1 1 0 0 0 1
1 0 2 1 0 0
0 2 5 1 1 0
0 1 2 1 0 1
replace row 3 by 1 3R R
1 0 2 1 0 0
0 2 5 1 1 0
0 1 2 1 0 1
2 3R R
1 0 2 1 0 0
0 1 2 1 0 1
0 2 5 1 1 0
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Example: Finding the Multiplicative Inverse of a 3 x 3 Matrix (continued)
Step 2 (cont)
1 0 2 1 0 0
0 1 2 1 0 1
0 2 5 1 1 0
1 0 2 1 0 0
0 1 2 1 0 1
0 2 5 1 1 0
replace row 2 by 2R
1 0 2 1 0 0
0 1 2 1 0 1
0 2 5 1 1 0
1 0 2 1 0 0
0 1 2 1 0 1
0 0 1 1 1 2
replace row 3 by 2 32R R
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Example: Finding the Multiplicative Inverse of a 3 x 3 Matrix (continued)
Step 2 (cont)
1 0 2 1 0 0
0 1 2 1 0 1
0 0 1 1 1 2
1 0 0 3 2 4
0 1 2 1 0 1
0 0 1 1 1 2
replace row 1 by 3 12R R
1 0 0 3 2 4
0 1 2 1 0 1
0 0 1 1 1 2
replace row 2 by 3 22R R
1 0 0 3 2 4
0 1 0 3 2 5
0 0 1 1 1 2
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Example: Finding the Multiplicative Inverse of a 3 x 3 Matrix (continued)
Step 2 Perform row operations on to obtain a matrix of the form
The result is
Step 3 Matrix B is A–1
3A I 3 .I B
1 0 0 3 2 4
0 1 0 3 2 5
0 0 1 1 1 2
1
3 2 4
3 2 5
1 1 2
A
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Example: Finding the Multiplicative Inverse of a 3 x 3 Matrix (continued)
Step 4 Verify the result by showing that AA–1 = I3 and A–1A = I3.
1
1 0 2 3 2 4
1 2 3 3 2 5
1 1 0 1 1 2
AA
3
1 0 0
0 1 0
0 0 1
I
1
3 2 4 1 0 2
3 2 5 1 2 3
1 1 2 1 1 0
A A
3
1 0 0
0 1 0
0 0 1
I
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Summary: Finding Multiplicative Inverses for Invertible Matrices
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Solving Systems of Equations Using Multiplicative Inverses of Matrices
The matrix equation
is abbreviated AX = B, where A is the coefficient matrix of the system and X and B are matrices containing one column, called column matrices. The matrix B is called the constant matrix.
1 1 1 1
2 2 2 2
3 3 3 3
a b c x d
a b c y d
a b c z d
A X B
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Solving a System Using A–1
If AX = B has a unique solution, then X = A–1B. To solve a linear system of equations, multiply A–1 and B to find X.
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Example: Using the Inverse of a Matrix to Solve a System
Solve the system by using A–1, the inverse of the coefficient matrix that was found in the previous example.
The linear system can be written as
2 6
2 3 5
6
x z
x y z
x y
1 0 2 6
1 2 3 5
1 1 0 6
x
y
z
A X B
The solution is given by1X A B
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Example: Using the Inverse of a Matrix to Solve a System (continued)
Solve the system by using A–1, the inverse of the coefficient matrix that was found in the previous example.
2 6
2 3 5
6
x z
x y z
x y
1
3 2 4 6
3 2 5 5
1 1 2 6
X A B
3(6) ( 2)( 5) ( 4)(6)
3(6) ( 2)( 5) ( 5)(6)
( 1)(6) 1( 5) 2(6)
4
2
1
4, 2, 1x y z
The solution setis {(4, –2, 1)}.
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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 message is then converted into a matrix. 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.
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Encoding a Word or Message
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Example: Encoding a Word
Use the coding matrix to encode the word BASE.
Step 1 Express the word numerically.
The numerical equivalent of BASE is 2,1,19,5.
Step 2 List the numbers in step 1 by columns and form a square matrix.
2 3
3 4
2 19
1 5
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Example: Encoding a Word (continued)
Use the coding matrix to encode the word BASE.
Step 3 Multiply the matrix in step 2 by the coding matrix.
2 3
3 4
2 3 2 19
3 4 1 5
( 2)(2) ( 3)(1) 2(19) ( 3)(5)
3(2) 4(1) 3(19) 4(5)
7 53
10 77
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Example: Encoding a Word (continued)
Use the coding matrix to encode the word BASE.
Step 3 Multiply the matrix in step 2 by the coding matrix.
The result is
Step 4 Use the numbers by columns, from the coded matrix in step 3 to write the encoded message.
The encoded message is –7, 10, –53, 77.
2 3
3 4
7 53
10 77
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Decoding a Word or Message That Was Encoded
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Example: Decoding a Word
Decode –7, 10, –53, 77.
Step 1 Find the inverse of the coding matrix.
2 3
3 4
a b
c d
1 1 d bA
ad bc c a
4 31( 2)(4) ( 3)(3) 3 2
4 311 3 2
4 3
3 2
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Example: Decoding a Word (continued)
Decode –7, 10, –53, 77.
Step 2 Multiply the multiplicative inverse of the coding matrix and the coded matrix.
4 3 7 53
3 2 10 77
4( 7) 3(10) 4( 53) 3(77)
( 3)( 7) ( 2)(10) ( 3)( 53) ( 2)(77)
2 19
1 5
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Example: Decoding a Word (continued)
Decode –7, 10, –53, 77.
Step 3 Express the numbers, by columns, from the matrix in step 2 as letters.
The numbers are 2, 1, 19, 5.
The decoded message is BASE.
2 19
1 5
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