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Chapter 3 Linear Algebra (continued)

Page history last edited by Chung 13 years, 9 months ago

Linear Algebra (continued)

 

1 Inverses of matrix

 

Let A be an nFormulan (square matrix). The inverses of A, denoted by Formula, is an n n matrix such that

                                                       Formula

If A has an inverse, then A is called a non-singular matrix and said to be invertible.

If A has no inverse, then A is called a singular matrix.

 

1.1 Existence and Uniqueness of an inverse

If the inverse exists, then the inverse is unique.

The inverse Formula of an nFormulan matrix A exists if and only if det(A)Formula0, and

                                                  Formula

In particular, the inverse of 2Formula2 matrix A is

                                                  Formula

Example 1.1. Find the inverse of the matrices

Formula

Solution.

 Formula

Formula

Since

Formula

Formula

 

Exercise 1.1. Find the inverse of

                                                  Formula

 

1.2 Using Gauss-Jordan Elimination to find inverses

 

The formula (1) is reasonable for inverting 3Formula3 matrices by hand. For large matrices, a computational procedure based on Gauss-Jordan elimination is more ecient. This procedure proceeds as follows. To nd the inverse of an invertible matrix A, we must find

Formula

We want to reduce A to the identity matrix by row operations and simultaneously apply these operations to I to produce Formula. To accomplish that we shall adjoin the identity matrix to the right side of A, thereby producing a matrix of the form

                                                  Formula

Then we shall apply row operations to this matrix until the left side is reduced to I. These operations will convert the right side to Formula, so that the nal matrix will have the form

                                                  Formula

This procedure works because the elementary row operations can be represented by a martix transformation, such that

                                                  Formula

then

                                                  Formula

where Formula

 

Example 1.2. Find the inverses of

                                             Formula

Solution.

Following the procedure, the computations are as follow

     Formula

     Formula

     Formula

     Formula

     Formula

     Formula

Thus,

                                             Formula

 

Exercise 1.2. Find the inverse of A in exercise 1.1 by Gauss-Jordon elimination.

 

 

2 General properties of inverses

 

2.1 Inverses of special matrices

 

Formula

 

2.2 Further properties of inverses

 

Suppose that A and B are invertible nFormulan matrices and k is any scalar, then

Formula

 

3 Solving system of linear equations by matrix inversion

 

If A is an invertible nFormulan matrix, then the system of equations Ax = b has exactly one solution, namely, Formula.

 

Example 3. Consider the system of linear equations

                                             Formula

 

In matrix form this system can be written as Ax = b, where

                                                  Formula

A is invertible and

                                                  Formula

Then the solution of the system is

                                                  Formula

or Formula

 

Exercise 3. Solve

                                             Formula

 

 

 

4 Powers of a matrix

 

If A is a square matrix, then we dene the nonnegative integer powers of A to be

                                             Formula

 

If A is invertible, then we dene the negative integer powers to be

                                             Formula

 

Because this denition parallels that for real numbers, the usual laws of exponents hold, i.e. for any integers r and s

                                             Formula

 

Example 4. Let

                                             Formula

Then

                                             Formula

                      Formula

Exercise 4. Consider a matrix

                                             Formula

Find Formula and Formula.

 

 

 

5 Matrix polynomial

 

If A is an nFormulan matrix, and if

                                             Formula

is any polynomial. By substituting A for x, then we have a matrix polynomial in A as

                                        Formula

where I is an nFormulan identity matrix.

 

Example 5. If Formula and

                                             Formula

then

Formula

 

Exercise 5. Consider Formula and A in exercise 4, Find p(A).

 

 

 

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