Download Chapter 3: The Inverse

Chapter 3
The Inverse
3.1 Introduction
Definition 1: The inverse of an nn matrix A is an nn matrix B having the
property that
AB = BA = I
B is called the inverse of A and is usually denoted by A-1 .
If a square matrix has an inverse, it is said to be invertible or nonsingular.
If it doesn’t possess an inverse, it is said to be singular.
Example: The inverse of
1 1
0 1


is
1  1 because
0 1 


1 1 1  1 1  1 1 1 1 0
0 1 0 1   0 1  0 1  0 1


 

 

3.1 Introduction: elementary matrices
An elementary matrix E is a square matrix that generates an elementary row
operation on a matrix A (which need not to be square) under the
multiplication EA.
We will need elementary matrices for finding inverses.
i)
To construct an elementary matrix that interchanges the ith row and the
jth row, begin with an identity matrix. First interchange the unity element
in the i-i position with the zero in j-i position, and then interchange the
unity element in the j-j position with the zero in i-j position.
ii) To construct an elementary matrix that multiplies the ith row of a matrix
by the nonzero scalar k, replace the unity element in the i-i poistion of the
identity matrix of appropriate order with the scalar k.
iii) To construct an elementary matrix that adds to the jth row ofa matrix k
times the ith row, replace the zero element in the j-i position of the identity
matrix with the scalar k.
Examples on the board.
3.1 Introduction: inverse of a matrix in
block diagonal form
If A can be partitioned into the block-diagonal form,  A1


A


 0
A2
A3
0






An 
Then A is invertible if and only if each of the diagonal blocks A1, …, An is
invertible and
 A 1
0 


A1  


 0

1
Examples on the board.
A2
1
A3
1






1
An 
3.2 Calculating Inverses
Theorem 1: A square matrix has an inverse if and only if reduction to row-reduced
form by elementary row operations results in a matrix having all unity elements on
the main diagonal.
Theorem 2: An nn matrix has an inverse if and only if it has rank n.
Theorem 1 suggests a procedure for finding the inverse. The idea is to
- first transform the matrix to a row-reduced form
- then keep applying elementary row operation (E3) to reduce it further to the identity
matrix.
Recall from Section 3.1 that these transformations can be accomplished by
multiplying the matrix with elementary matrices:
Ek Ek-1 … E3 E2 E1 A = I
But then A-1 = Ek Ek-1 … E3 E2 E1
Thus, to calculate the inverse we need to keep a record of the elementary matrices.
This is accomplished by applying the same elementary operations to both A and an
identity matrix since (Ek Ek-1 … E3 E2 E1)I = Ek Ek-1 … E3 E2 E1 = A-1
Example 1:
Example 2:
A Non-Invertible Matrix
3.3 Simultaneous Equations
A linear system can be solved using inverses.
Suppose we have a linear system
Ax=B
where A is invertible. Then
A-1 A x = A-1 B
I x = A-1 B
x = A-1 B
Theorem: If A is invertible, then A x = B has one and only one
solution.
Examples on the board.
3.4 Properties of the Inverse
•
(A-1)-1 = A
• (AB)-1 = B-1 A-1
• (AT)-1 = (A-1)T
• (λA)-1 = (1/ λ) A-1
• The inverse of a matrix is unique.
• The inverse of a nonsingular symmetric matrix is symmetric.
•
The inverse of a nonsingular upper or lower triangular matrix
is again upper or lower triangular matrix respectively.
• If A is nonsingular then we define A-n = (A-1)n