3_matrix.ppt

8/11/2019 3_matrix.ppt

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Matrix

MSc. Do Thi Phuong Thao

Fall 2013


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Learning resource

Kenneth Hoffman, Linear Algebra, 2nd

Edition, Prentice Hall


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Contents

Definition

Elementary row operations

Row-Reduced Echelon Matrices

Matrix Algebra

Inverse matrix


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Definition

A = [aij]mxn =

amnamam

naaa

naaa

...21

............

2...2221

1...1211


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Examples

If m=n: A is square matrix

annanan

naaa

naaa

...21

............

2...2221

1...1211


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Examples

Identity matrix nxn:

All elements in main diagonal = 1

All other elements = 0

1...00

............

0...10

0...01

In =


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Examples

Row matrix: m=1

Column matrix: n=1

bm

b

b

anaa

...

2

1

...21


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Examples

Triangle matrix:

ann

naa

naaa

...00

............

2...220

1...1211

annanan

aa

a

...21

............

0...2221

0...011


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Examples

Row-reduced echelon matrix :

All zero rows are at the bottom of the matrix

The leading entry in any nonzero row is 1.

If the leading entry of row r1 is in the column c1,and the leading entry of row r2 is in the column

c2, and if r1


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Example of Row-reduced echelon matrix


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Elementary Row Operations

Three elementary row operations that can be

performed on matrix A are:

Multiplication of one row by a non-zero scalar

Add a scalar multiple of one row to another row

Interchange of two rows


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Examples


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Theorem

To each elementary row operation ethere

corresponds an elementary row operation e,

of the same type as e, such that e(e(A)) =

e(e(A)) =Afor eachA. In other words, the inverse operation

(function) of an elementary row operation

exists and is an elementary row operation ofthe same type.


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Examples

Multiplication of one row by a non-zero scalarc Multiplication of one row by a non-zero scalar c-1

Replacement of row rby row rplus ctimesrow s Replacement of row rby row rplus (- c)times rows

Interchange of two rows rand s Interchange of two rows sand r


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Examples


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Contents

Definition



Matrix Algebra

Inverse matrix


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Example


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Row-reduced echelon matrix

1. All zero rows are at the bottom of the matrix

2. The leading entry in any nonzero row is 1.

3. If the leading entry of row r1 is in the column

c1, and the leading entry of row r2 is in the

column c2, and if r1


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Theorem

Every mx nmatrixAis row-equivalent to a

row-reduced echelon matrix.


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Example 1

Find a row-reduced echelon matrix which is

equivalent to B:


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Example 1


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Example 1


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Matrix algebra

A = [aij]mxn =

B= [bij]mxn =

amnamam

naaa

naaa

...21

............

2...2221

1...1211

bmnbmbm

nbbb

nbbb

...21

............

2...2221

1...1211


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Matrix addition

A = [aij]mxn, B = [bij]mxn:

A+B = [aij+bij]mxn

A + B = B + A


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Multiplication of a scalar and matrix

A = [aij]mxn, k,h are real number:

kA = [kaij]mxn

k(A+B) = kA+kB

(k+h)A = kA + hA

k(hA) = (kh)A


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Multiplication of Matrices

A = [aij]mxp, B = [bij]pxn

C = AB = [cij]mxn:

cij = ai1.b1j+ai2.b2j++aip.bpj

cij =

AB and BA are different.

A(B+C) = AB + AC

(B+C)A = BA +ACA(BC) = (AB)C

k(BC) = (kB)C = B(kC)

p

k

bkjaik1

.


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Contents

Definition



Matrix Algebra

Inverse matrix


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Inverse matrix

Definition:

An nxn identity matrix In =

Let A is an nxn matrix. Then B is an inverse of A

if:

AB = BA = In

so B must also be nxn and B = A-1

InA = A In= A

1...00

............

0...10

0...01


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Examples


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Theorem

Definition:

A square matrix is said to be nonsingular or

invertible if it has an inverse.

If it has no inverse, the matrix is singular.

Theorem:

An nxn matrix is nonsingular if and only if:

AR= In


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A method for finding A-1

(Gauss-Jordan method)


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A method for finding A-1


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Thank you for your attention.

Thats all for today.

3_matrix.ppt

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