Section 4 Inverse Matrix



Section 4 Inverse Matrix

4.1 Definition:

Definition of inverse matrix:

An [pic] matrix A is called nonsingular or invertible if there exists an [pic] matrix B such that

[pic],

where [pic] is a [pic] identity matrix. The matrix B is called an inverse of A. If there exists no such matrix B, then A is called singular or noninvertible.

Theorem:

If A is an invertible matrix, then its inverse is unique.

[proof:]

Suppose B and C are inverses of A. Then,

[pic].

◆

Note:

Since the inverse of a nonsingular matrix A is unique, we denoted the inverse of A as [pic].

Note:

If A is not a square matrix, then

• there might be more than one matrix L such that

[pic].

• there might be some matrix U such that

[pic]

Example:

Let

[pic].

Then,

• there are infinite number of matrices L such that [pic], for example

[pic] or [pic].

• As [pic],

[pic] but [pic].

4.2 Calculation of Inverse Matrix:

1. Using Gauss-Jordan reduction:

The procedure for computing the inverse of a [pic] matrix A:

1. Form the [pic] augmented matrix

[pic]

and transform the augmented matrix to the matrix

[pic]

in reduced row echelon form via elementary row operations.

2. If

(a) [pic], then [pic].

(b) [pic], then [pic] is singular and [pic] does not exist.

Example:

To find the inverse of [pic], we can employ the procedure introduced above.

1.

[pic].

[pic] [pic]

[pic] [pic]

[pic] [pic]

[pic] [pic]

2. The inverse of A is

[pic].

Example:

Find the inverse of [pic]if it exists.

[solution:]

1. Form the augmented matrix

[pic].

And the transformed matrix in reduced row echelon form is

[pic]

2. The inverse of A is

[pic].

Example:

Find the inverse of [pic]if it exists.

[solution:]

1. Form the augmented matrix

[pic].

And the transformed matrix in reduced row echelon form is

[pic]

2. A is singular!!

2. Using the adjoint [pic] of a matrix:

As [pic], then

[pic].

Note: [pic] is always true.

Note: As [pic] [pic] A is nonsingular.

4.3 Properties of The Inverse Matrix:

The inverse matrix of an [pic] nonsingular matrix A has the following important properties:

1. [pic].

2. [pic]

3. If A is symmetric, So is its inverse.

4. [pic]

5. If C is an invertible matrix, then

a. [pic]

b. [pic].

6. As [pic]exists, then

[pic].

[proof of 2]

[pic]

similarly,

[pic].

[proof of 3:]

By property 2,

[pic].

[proof of 4:]

[pic].

Similarly,

[pic].

[proof of 5:]

Multiplied by the inverse of C, then

[pic].

Similarly,

[pic].

[proof of 6:]

[pic]

[pic].

Multiplied by [pic] on both sides, we have

[pic].

[pic]

can be obtained by using similar procedure.

Example:

Prove that [pic].

[proof:]

[pic]

Similar procedure can be used to obtain

[pic]

4.4 Left and Right Inverses:

Definition of left inverse:

For a matrix A,

[pic],

with more than one such L. Then, the matrices L are called left inverse of A.

Definition of right inverse:

For a matrix A,

[pic],

with more than one such R. Then, the matrices R are called left inverse of A.

Theorem:

A [pic] matrix [pic] has left inverses only if [pic].

[proof:]

We prove that a contradictory result can be obtained as [pic]and [pic] having a left inverse. For [pic], let

[pic]

Then, suppose

[pic]

is the left inverse of [pic]. Then,

[pic].

Thus,

[pic]

Since [pic]and both M and X are square matrices, then [pic].

Therefore,

[pic].

However,

[pic].

It is contradictory. Therefore, as [pic], [pic] has no left inverse.

Theorem:

A [pic] matrix [pic] has right inverses only if [pic].

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