Chapter 3The Inverse

3.1 Introduction

Definition 1: The inverse of an nn matrix A is an nn 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 is because

10

11

10

11

10

01

10

11

10

11

10

11

10

11

3.1 Introduction: elementary matricesAn 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,

nA

A

A

A

A

0

0

3

2

1

Then A is invertible if and only if each of the diagonal blocks A1, …, An is invertible and

1

13

12

11

1

0

0

nA

A

A

A

A

Examples on the board.

3.2 Calculating InversesTheorem 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 nn 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 EquationsA linear system can be solved using inverses.

Suppose we have a linear system

A x = 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