Chapter_1-Slides(Sem 1 2013)

7/27/2019 Chapter_1-Slides(Sem 1 2013)

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MATB 314 & MATB 253 - LINEAR ALGEBRA

2009 Bulletin Data: Systems of Linear Equations and Matrices,

Determinants, Euclidean Vector Spaces,

General Vector Spaces, Inner Product

Spaces, Eigenvalues and Eigenvectors,

Applications.

Textbook: Anton H. and Rorres C.: Elementary Linear

Algebra (Applications Version), 9 th Edition,

John Wiley & Sons, Inc, 2005.

Objectives:

At the end of the course, students should be able to solve systems of linear equations using the Gaussian/ Gauss-Jordan elimination, Cramers rule and theinverse of a matrix, calculate the determinants, find the standard matrix of linear transformations from R n to R m , determine whether a set of objects together withoperations defined on it form a vector space, test for a subspace, show whether aset of vectors is a basis, determine the dimension of a vector space, find a basis for the row space, column space and nullspace of a matrix, calculate the rank andnullity of a matrix, give examples of inner product spaces, use the Gram- Schmidt

process to find an orthonormal basis, find the eigenvalues and the correspondingeigenvectors of a square matrix, how to diagonalize a matrix. Some applications of linear algebra to engineering are discussed.


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Chapter 1

SYSTEMS OF LINEAR EQUATIONS AND MATRICES

Contents

Contents..................................................................................................................... 2

1.1 Introduction to Systems of Linear Equations ....................................................3

................................................................................................................................ 3

1.2 Matrices and Matrix Operations........................................................................ 10

1.3 Gaussian Elimination ...................................................................................... 18

____________________________________________________................................................29

Hence,...................................................................................................................... 29

Recall :......................................................................................................................29

1.4 Properties of Matrix Operations........................................................................29

Let ........................................................................................................................... 36

1.5 Elementary Matrices and a Method for Finding ...............................................36

1.6 Further Results on Systems of Equations and ..............................................48

Invertibility....................................................................................................... 481.7 Application to Electrical Networks ..................................................................54

1.8 Diagonal and Triangular Matrices .................................................................59

___________________________________________________ _


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Introduction to Linear Algebra

SYSTEMS OF LINEAR EQUATIONS AND MATRICES

1.1 Introduction to Systems of Linear Equations

Linear algebra is the branch of mathematics concerned with thestudy of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations in finitedimensions. Vector spaces are a central theme in modernmathematics; thus, linear algebra is widely used in both abstractalgebra and functional analysis. Linear algebra also has aconcrete representation in analytic geometry and it isgeneralized in operator theory. It has extensive applications inthe natural sciences and the social sciences, since nonlinear models can often be approximated by a linear model.


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In mathematics and linear algebra, a system of linear equationsis a set of linear equations such as

1 2 3

1 2 3

1 2 3

4 6

3 2 7

2 2 3 3

x x x

x x x

x x x

+ = + =+ =

A standard problem is to decide if any assignment of values for the unknowns can satisfy all three equations simultaneously, andto find such an assignment if it exists. The existence of asolution depends on the equations, and also on the availablevalues (whether integers, real numbers , and so on).

There are many different ways to solve systems of linear

equations, such as substitution, elimination, matrix anddeterminants. However, one of the most efficient ways isgiven by Gaussian elimination (matrix).

In general, a system with m linear equations and n unknownscan be written as

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

...

...

...

n n

n n

m m mn n m

a x a x a x b

a x a x a x b

a x a x a x b

+ + + =+ + + =

+ + + =

M M


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where 1 2, ,..., n x x x are the unknowns and the numbers

11 12, ,..., mna a a are the coefficients of the system.

We can collect the coefficients in a matrix as follows:

If we represent each matrix by a single letter, this becomes

[ ] [ ] [ ] A x b=where A is an mn matrix, x is a column vector with n entries,and b is a column vector with m entries.

Gauss-Jordan elimination applies to all these systems, even if the coefficients come from an arbitrary field.


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11 12 1 1 1

21 22 2 2 2

1 1

, ,

n

n

m m mn n m

a a a x b

a a a x b A x b

a a a x b

= = =

K

K M M L M M M

K

If the field is infinite (as in the case of the real or complexnumbers), then only the following three cases are possible(exactly one will be true)

For any given system of linear equations :

the system has no solution (the system is over determined)

the system has a single solution (the system is exactlydetermined)

the system has infinitely many solutions (the system isunderdetermined).

A system of equations that has at least one solution is calledconsistent ; if there is no solutions it is said to be inconsistent .


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___________________________________________________ _

A system of the form 0 Ax = is called a homogeneous systemof linear equations. The set of all solutions of such ahomogeneous system is called the nullspace of the matrix A.

Example 1 :

11 1 12 2 1

21 1 22 2 2

1 1 2 2

... 0

... 0

... 0

n n

n n

m m mn n

a x a x a x

a x a x a x

a x a x a x

+ + + =+ + + =

+ + + =M M

If the system is homogeneous and 1 2 ... 0n x x x= = = = thenwe have a trivial solution.

If the system is homogeneous and at least one x i 0 , thenwe have a nontrivial solution.

Since a homogeneous linear system always has the trivialsolution; there are only 2 possibilities for its solution.

(a) The system has only the trivial solution


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(b) The system has infinitely many solutions in

addition to the trivial solution.

(Howard,2005)

___________________________________________________

_

Augmented Matrices

Linear Equations matrix form

Augmented Matrix

11 12 1 1

21 22 2 2

1 1

n

n

m m mn m

a a a b

a a a b

a a a b

K

K

M M L M M

K

TheoremA homogeneous system of linear equations with more unknownsthan equations has infinitely many solution.


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To find the solution(s) of a given system of linear equations, we

have to reduce an augmented matrix to row echelon form or

reduced row-echelon form . The reduction is done by using

operations on the rows of the augmented matrix. The operations

are called the elementary row operations.

_____________________________________________________________________________ _

Elementary Row Operation

An elementary row operations (ERO) on a matrix A is one of thefollowing :

1. Multiply a row by a nonzero constant, c cR i R i

2. Switching any two rows R i R j

3. Add c times of one row to another row cR i + R j R j

Elementary row operations are used to reduce an augmentedmatrix or matrix to row echelon form or reduced row-echelonform . Reducing the matrix to row echelon form is calledGaussian elimination and to reduced row-echelon form is called GaussJordan elimination . Gaussian elimination is anefficient algorithm for solving systems of linear equations. An


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extension of this algorithm, GaussJordan elimination, reducesthe matrix further to reduced row echelon form.

In mathematics , GaussJordan elimination is a version of Gaussian elimination that puts zeros both above and below each pivot element as it goes from the top row of the givenmatrix to the bottom. (http://en.wikipedia.org )

_____________________________________________________________________________ _

1.2 Matrices and Matrix Operations

Matrix

Matrix is a rectangular array of numbers or, moregenerally, a table consisting of abstract quantities that can

be added and multiplied .

The numbers in the array are called the entries in the matrix Matrices are used to describe linear equations , keep track of

the coefficients of linear transformations and to record datathat depend on two parameters.
http://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/w/index.php?title=Pivot_element&action=edithttp://en.wikipedia.org/w/index.php?title=Pivot_element&action=edithttp://en.wikipedia.org/http://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Ring_(mathematics)http://en.wikipedia.org/wiki/Ring_(mathematics)http://en.wikipedia.org/wiki/System_of_linear_equationshttp://en.wikipedia.org/wiki/Coefficienthttp://en.wikipedia.org/wiki/Linear_transformationhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/w/index.php?title=Pivot_element&action=edithttp://en.wikipedia.org/http://en.wikipedia.org/wiki/Numberhttp://en.wikipedia.org/wiki/Ring_(mathematics)http://en.wikipedia.org/wiki/Ring_(mathematics)http://en.wikipedia.org/wiki/System_of_linear_equationshttp://en.wikipedia.org/wiki/Coefficienthttp://en.wikipedia.org/wiki/Linear_transformation


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Matrices can be added, multiplied, and decomposed invarious ways, making them a key concept in linear algebra and matrix theory .

Definitions and notations

The horizontal lines in a matrix are called rows and thevertical lines are called columns.

A matrix with m rows and n columns is called an m-by- nmatrix (written mn) and m and n are called its dimensions.

The dimensions of a matrix are always given with thenumber of rows first, then the number of columns.

The entry of a matrix A that lies in the i -th row and the j-thcolumn is called the i,j entry or ( i, j)-th entry of A. This iswritten as a ij or ( A)ij. The row is always noted first , thenthe column.

We often write to define an mn matrix A with eachentry in the matrix [ a ij]mxn called a ij for all 1 i m and

1 j n
http://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Matrix_theoryhttp://en.wikipedia.org/wiki/Linear_algebrahttp://en.wikipedia.org/wiki/Matrix_theory


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(Adapted from http://en.wikipedia.org )

A matrix where one of the dimensions equals one is oftencalled a vector , and interpreted as an element of realcoordinate space.

A 1 n matrix (one row and n columns) is called a rowvector , and an m 1 matrix (one column and m rows) iscalled a column vector .

Example 8 :
http://en.wikipedia.org/http://en.wikipedia.org/


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[ ]

[ ]

12 4 1 3

0or 1

21 0 2 5 0

4

Row vector Column vector

A square matrix is a matrix which has the same number of rows and columns.

A square matrix of order n and the entries a 11, a 22, . . . , a nn arethe main diagonal of A.

The unit matrix or identity matrix I n, with elements onthe main diagonal set to 1 and all other elements set to 0,satisfies M I n=M and I n N=N for any m-by- n matrix M andn -by- k matrix N .

Example 9 :

if n = 3: I 3 =

1 0 0

0 1 0

0 0 1

__________________________________________________________

Operations on MatricesDefinitionTwo matrices are defined to be equal if they have the samesize and their corresponding entries are equal .


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In matrix notation, if A= [ a ij] and B = [ b ij] have the same size,then A = B if and only if ( A)ij= ( B)ij

Example 10

0 2 1 2

3 1 1 82 1 5 5

a c

b d e f

=

1, 8, 0, 3, 2 1a b c d e and f = = = = = =

__________________________________________________________

Addition and Subtraction ( Only on same size matrices )

Given m-by- n matrices A and B ,their sum A + B is the m-by- n matrix computed by addingcorresponding entries

( A + B )ij = ( A) ij + ( B) ij

= a ij + b ij their difference A B is the m-by- n matrix computed bysubtracting corresponding entries ( A - B)ij = ( A) ij - ( B) ij

= a ij - b ij Example 11 :


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1. Addition

1 0 2 1 2 3 2 2 1

3 1 3 4 5 6 7 6 32 1 5 7 8 9 5 7 14

+ =

2. Subtraction

1 0 2 1 2 3 0 2 5

3 1 3 4 5 6 1 4 92 1 5 7 8 9 9 9 4

=

_____________________________________________________________________________ _

Scalar Multiples

If A is any matrix and c is any scalar, then the product cA is thematrix obtained by multiplying each entry of the matrix A byc. The matrix cA is said to be a scalar multiple of A. In matrixnotation, if A = a ij , then ( cA) ij = c(A) ij = ca ij

Example 12 :

Linear combination c1A1 + c2A2 + . . . + cnAn

1 0 1 2 4 3 1 0 0

3 2 3 0 2 1 3 0 5 3 2 1

0 4 1 0 2 6 5 4 0

+

6 8 9

7 5 5

25 28 15

=

Multiplying Matrices


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Multiplication of two matrices is well-defined only if thenumber of columns of the left matrix is the same as thenumber of rows of the right matrix.

If A is an m-by- n matrix and B is an n -by- p matrix, thentheir matrix product AB is the m- by - p matrix ( m rows, pcolumns) given by

Example 13

2 1 (2)(3) ( 1)(1) (2)(1) ( 1)(2)3 1

( ) 0 2 (0)(3) (2)(1) (0)(1) (2)(2)1 2

1 3 (1)(3) (1)( 3) (1)(1) ( 3)(2)

5 02 4

0 5

i

+ + = + + + +

=

______________________________________________

( )ii Find the (2,3) th in the following matrix multiplication :


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(2,3)

4 1 4 3 _ _ _ _ 1 2 40 1 3 1 _ _ _ _ 2 6 02 1 5 2

=

(2,3) th entry : [ ]4

2 6 0 3 8 (18) 0 26

5

= + + =

Matrix Products as Linear Combination

11 12 1 1

21 22 2 2

1 1

,

n

n

m m mn n

a a a x

a a a x A x

a a a x

= =

K

K

M M L M M

K

Then

11 12 1

21 22 21 2

1 1

n

nn

m m mn

a a a

a a a Ax x x x

a a a

= + + +

LM M M

11 1 12 2 1

21 1 22 2 2

1 1 2 2

...

...

...

n n

n n

m m mn n

a x a x a x

a x a x a x

a x a x a x

+ + ++ + +

=

+ + +M M M

_____________________________________________________________________________


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1.3 Gaussian Elimination

Echelon Forms

Reducing the augmented matrix of a system to row-echelonform

To be in this form, a matrix must have the following properties:

1. If there are any rows that consist entirely of zeros, then theyare grouped together at the bottom of the matrix.

2. If a row does not consist entirely of zeroes, then the firstnonzero number in the row is 1, We call this a leading 1.

3. If any two successive rows that do not consist entirelyzeros, the leading 1 in the lower row occurs farther to theright than the leading 1 in the higher row.

4. In Reduced Row-Echelon , each column that contains aleading 1 has zeros everywhere else in the higher row.

____________________________________________________________________________

Row-Echelon

Example 2 :

01 0 3

0 1 1 2

0 0 11


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0 1 2 3 2

70 0 1 120 0 0 100 0 0 0

,

1 4 6 3 1

30 0 1 350 0 0 100 0 0 0

_____________________________________________________________________________ _

Reduced Row-Echelon

Example 3 :

1 0 0 3

0 1 0 2

0 0 1 1

,

0 1 0 0 2

0 0 1 0 0

0 0 0 1 2

0 0 0 0 0

,

1 0 0 0 2

0 1 0 0 3

0 0 1 0 2

0 0 0 1 1

___________________________________________________ _

The augmented matrix for a system of linear equations is put inreduced row-echelon form, and then the solution set of thesystem will be evident by inspection or after a few simple steps.

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the givenreduced row-echelon form.


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Example 4:

(a)

1 0 0 50 1 0 2

0 0 1 4

means by

1 2 35 , 2 , 4 x x x= = =

(b)

1 0 0 4 1

0 1 0 2 6

0 0 1 3 2

means by

3 4 2 4 1 43 2 , 2 6 , 4 1 x x x x x x+ = + = + =

(c)

1 6 0 0 4 2

0 0 1 0 3 1

0 0 0 1 5 2

0 0 0 0 0 0

use first 3 rows to solve the system of linear equation

(d) 1 0 0 3

0 1 0 2

0 0 0 4

means no solution


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___________________________________________________ _

Solutions of Linear Systems

Elimination Methods : Gaussian Elimination andBack Substitution

Example 5 :

Solve the system by Gaussian Elimination

1.1 2 3

1 2 3

1 3

2 3 5

2 5 3 3

8 17

x x x

x x x

x x

+ + =+ + =

+ =

Solution

Gaussian Elimination

1 2 2

1 3 3 2 3 3

1 2 3 5 2 1 2 3 5 1 2 3 5

2 5 3 3 0 1 3 7 0 1 3 7

1 0 8 17 0 2 5 12 2 0 0 1 2

R R R

R R R R R R

+ + +

3 3

1 2 3 5

0 1 3 7( 1)

0 0 1 2 R R

Back Substitution

1 2 3 2 3 32 3 5 , 3 7 , 2 x x x x x x+ + = = =


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( )( ) ( ) ( ) ( )

2 2

1 1

3 2 7 7 6 1

2 1 3 2 5 5 2 6 1

x x

x x

= = + =

+ + = = + =

1 2 31 , 1 , 2 x x x = = =

_____________________________________________________________________________ _

2.1 2 3

1 2 3

1 2 3

2 2 2 4

3 2 3

4 3 2 3

x x x

x x x

x x x

+ = + =

+ =

Solution

1 1

2 2 2 4 1 1 1 21

3 2 1 3 3 2 1 324 3 2 3 4 3 2 3

R R


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1 2 2

1 3 3

2 2

1 1 1 23 1 1 1 2

4 30 5 4 3 0 1

5 54 0 7 6 5 1 0 7 6 5

5

R R R

R R R R R

+ +

3 32 3 3

51 1 1 2 1 1 1 2

7 24 3 4 3

0 1 0 15 5 5 52 4 0 0 1 2

0 05 5

R R R R R

+

1 2 3 2 3 3

4 31 2 , , 2

5 5 x x x x x x+ = = =

( )

( ) ( ) ( ) ( )

2 2

1 1

4 3 3 8 52 , 1

5 5 5 5 5

1 2 2 , 2 1 2 1

x x

x x

= = + = =

+ = = + =

1 2 31 , 1 , 2 x x x = = =

Gauss - Jordan Elimination

Example 6 :

(a) Consider the linear system1 2 3

1 2 3

1 3

2 3 5

2 5 3 3

8 17

x x x

x x x

x x

+ + =+ + =

+ =.


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Solve the system using Gauss-Jordan elimination method.

Solution (a)

Gauss - Jordan Elimination

1 2 2 2 1 1

1 3 3 2 3 3

1 2 3 5 2 1 2 3 5 2 1 0 9 19

2 5 3 3 0 1 3 7 0 1 3 71 0 8 17 0 2 5 12 2 0 0 1 2

R R R R R R

R R R R R R

+ +

+ +

3 1 1

3 33 2 2

1 0 9 19 9 1 0 0 1

0 1 3 7 0 1 0 1( 1)

0 0 1 2 3 0 0 1 2

R R R

R R R R R

+ +

1 2 31 , 1 , 2 x x x = = =

(b) Consider

1 2 3

1 2 3

1 2 3

2 2 2 4

3 2 3

4 3 2 3

x x x

x x x

x x x

+ = + =

+ =.

Solve the system using Gauss-Jordan elimination method.

Solution (b)Gauss-Jordan elimination


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1 1

2 2 2 4 1 1 1 213 2 1 3 3 2 1 324 3 2 3 4 3 2 3

R R

1 2 2

1 3 3

1 1 1 2 3 1 1 1 2

3 2 1 3 0 5 4 3

4 3 2 3 4 0 7 6 5

R R R

R R R

+ +

2 21 1 1 1 25

4 30 1

5 50 7 6 5

R R

( ) 2 1 1

2 3 3

1 71 0 5 514 3

0 15 5

7 2 40 0

5 5

R R R

R R R

+ +


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3 1 1

3 2 23 3

1 7 11 05 5 1 0 0 154 3

0 1 0 1 0 15 5 4 0 0 1 25 0 0 1 2

52

R R R

R R R R R

+ +

1 2 31 , 1 , 2 x x x = = =

Homogeneous Linear Systems

Example 7 :

(a) Solve

0

2 0

3 4 0

x y z

x y z

x y z

+ =+ + =+ =

Solution (a)


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1 2 2

1 3 3

2 3 3

0

2 03 4 0

1 1 1 0 1 1 1 01 2 1 0 0 1 2 0

3 4 1 0 3 0 1 2 0

1 1 1 00 1 2 00 0 0 0

x y z

x y z

x y z

R R R

R R R

R R R

many solutions

+ =+ + =

+ =

+

+

+

Since last row is entirely zero entries, the homogeneous systemhas nontrivial solution (many solutions )

( ) ( )

1 2 3 2 3

3

2 2

1 1

0 , 2 0

2 0 2

2 0 3

x x x x x

Let x t

x t x t

x t t x t

+ = + =

=

+ = =

+ = =

1 2 33 , 2 , x t x t x t = = =


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(b) Solve

2 5 0

3 2 2 0

4 4 5 0

x y z

x y z

x y z

+ = + =

+ =

Solution (b)

1 2 2

1 3 3

1 2 5 0 3 1 2 5 0

3 2 2 0 0 4 13 0

4 4 5 0 4 0 4 15 0

R R R

R R R

+

+

2 22 3 3

1 01 2 51 2 5 0 4

130 4 13 0 0 1 0

4

0 0 2 0 0 0 2 0

R R R R R

+

3 32 1 1

0 03 311 0 1 02 22 2

13 130 1 0 0 1 0

4 40 0 2 0 0 10 0

R R R R R

+

3 2 2

3 1 1

131 0 0 040 1 0 0

3 0 0 1 02

R R R

R R R

+ +


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Therefore the system has exactly one solution. ( Trivialsolution )

1 2 30 , 0 , 0 x x x = = =

___________________________________________________

_

Hence,

Recall :

Since a homogeneous linear system always has the trivialsolution; there are only 2 possibilities for its solution.

(a) The system has only the trivial solution

(b) The system has infinitely many solutions in

addition to the trivial solution.

(Howard,2005)

1.4 Properties of Matrix Operations

Properties of Matrix Arithmetic

Assuming that the sizes of the matrices are such that theindicated operations can be performed, the following rules of matrix arithmetic are valid. A, B ,C are matrices and a ,b, c are any constant

(a) A + B = B + A (Commutative law for addition)(b) A + (B + C) = (A + B) +C (Associative law for addition)


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(c) A(BC) = (AB)C (Associative law for multiplication)(d) A (B + C) = AB + AC (Left distributive law)(e) (B + C) A = BA + CA (Right distributive law)(f) A (B C) = AB AC(g) (B C) A = BA CA(h) a (B + C) = aB + aC(i) a (B C) = aB aC(j) (a + b)C = aC + bC(k) (a b)C = aC bC(l) a (bC) = ( ab ) C

(m) a (BC) = ( aB)C = B( aC) _____________________________________________________________________________ _

Zero Matrices

[ ]0 0 0

0 0 0 0 0 00 , , 0 0 0 ,

0 0 0 0 0 00 0 0

Properties of Zero Matrices

(a) A + 0 = 0 + A = A(b) A-A = 0(c) 0 A = A(d) A0 = 0 ; 0A =0

_____________________________________________________________________________ _


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Identity Matrices

2

3

4

1 0 0 01 0 0

1 0 0 1 0 0

, 0 1 0 ,0 1 0 0 1 00 0 1

0 0 0 1 I

I I

Inverse of A

Example 14

2 3 8 3Consider and .

5 8 5 2

Find and

A B

AB BA

= =

Definition

If A is a square matrix, and if a matrix B of the same size can be found such that AB = BA= I, then A is to be invertible andB is called an inverse of A . If no such matrix B can be found,then A is said to be singular .

Theorem

If R is the reduced row-echelon form of an nxn matrix A ,

then either R has a row of zeros or R is the identity matrix I n.


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Solution

2 3 8 3 1 0 5 8 5 2 0 1

8 3 2 3 1 0.

5 2 5 8 0 1

AB

BA

= =

= =

Note: A

1

= B and B

1

= A _________________________________________________________

Method of finding inverse of 2 x 2 invertible matrix

Properties of Inverses

1) If B and C are both inverses of the matrix A, then B = C

2) A A 1 = A 1A = I

3) If A and B are invertible matrices of the same size ,

Theorem

The matrix a b Ac d =

is invertible if 0ad bc , in

which case the inverse is given by the formula

1 1 - -=-

- -

d bd b ad bc ad bc Ac a c aad bc

ad bc ad bc

=


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then AB is invertible and (A B) 1 = B 1A1

_________________________________________________________________________________________

Power of a Matrix

Definition If A is a square matrix, then we define the nonnegative integer powersof A to be

a. A 0 = I

b. A n = A . A . A . . . A (n> 0)

n factors

c. A n = (A 1)n = A 1 . A1 . . . A1

n factors

Laws of Exponents

a . If A is a square matrix and r and s are integers, then

Ar A s = Ar+s


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b. ( Ar ) s = A r s

c. If A is an invertible matrix, then : A1 is invertible and

( A1 ) 1 = A

d. If A is an invertible matrix, then : An is invertible and

( An ) 1 = ( A1 )n for n = 1, 2,

e. For any nonzero scalar k , the matrix kA is invertible and

(kA)1 = (1/ k ) A1

__________________________________________________________________________________________

Transpose of a Matrix (AT)i j = (A) j i

Example 15 :

T T1 0 1 1 2 0 1 4

1 2 31. 2 3 0 0 3 4 2. 2 5

4 5 60 4 1 1 0 1 3 6

= =

DefinitionIf A is any m x n matrix, then the transpose of A, denoted by

A T, is defined to be the n x m matrix that the results frominterchanging the rows and columns of A ; that is, the firstcolumn of AT is the first row of A, the second column of AT

is the second row of A, and so forth.


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Properties of the Transpose

If the sizes of the matrices are such that the stated operations can be performed, then

(a) (( A )T)T = A

(b) ( A+ B)T= A T + B T and ( A B)T= A T B T

(c) (k A )T= k A T , where k is any scalar

(d) ( AB )T= B T A T

Invertibility of a Transpose

Example 16 :

Theorem

If A is an invertible matrix, then A T is also invertible and

( A T)1 =( A 1)T


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Let 12 3 8 3

and .5 8 5 2

A A = =

( )

( )

1

1 1

2 5 8 5 8 51= and = =

3 8 3 2 3 21

8 3 8 51and =

5 2 3 21

T T

T

A A

A A

= _____________________________________________________________________________________

1.5 Elementary Matrices and a Method for Finding 1 A

Recall :

Given a system of linear equations, then for the matrixequation Ax =b in which A is invertible n x n matrix for eachn x 1 matrix b, has a unique solution x = A1b

In this section we will develop a method for finding the inverseof a matrix, if it exists. To use this method, we do not have tofind out first whether A1 exists. We start to find A1 ; if in thecourse of the computation we hit a certain situation, then weknow that A1 does not exists.. Otherwise we proceed tothe endand obtain A1 . This method requires that the elementary roeoperations be performed on A.


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The Algorithm involved :If we place A and I n side-by-side to form an augmented matrix[ A | I n ], then row operations on this matrix produce identicaloperations on A and on I n. Then either there are row operationsthat transform A to I n and I n to A1 , or else A is not invertible.

1

A I

I A

Example :17

0 1 2

Given 1 0 3

4 3 8

A

=

0 1 2 1 0 0

1 0 3 0 1 0

4 3 8 0 0 1

1 0 3 0 1 00 1 2 1 0 0

4 3 8 0 0 1

A I


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1 0 3 0 1 0

0 1 2 1 0 0

0 3 4 0 4 1

1 0 3 0 1 0

0 1 2 1 0 0

0 0 2 3 4 1

1 0 3 0 1 0

0 1 2 1 0 0

0 0 1 3/ 2 2 1/ 3

___________________________________________________ _

Elementary Matrices

.DefinitionAn n x n matrix is called an elementary matrix if it can beobtained from the n x n identity matrix I n by performing a

single elementary row operation


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Example 18 :I E 1

1 3 3

1 0 0 1 0 00 1 0 0 1 0 .

0 0 1 1 0 1 R R R

I E +

= =

Row Operations and Inverse Row Operations

Rowoperation on Ithat producesE

Example Rowoperationon E thatreproducesI

Example

Multiply row i by c 0

[I] R 1 = 3R 1 [E] Multiplyrow i by 1/ c

1/3R 1[E] = [I] R 1

Interchangerows i and j

[I] 3 2 R R [E] Interchangerows i and j

[E] 2 3 R R [I]

Add c timesrow i to row j

[I] R 1 = R 1 +2R 3 [E] Add c times row i to row j

R 12R 3[E] =[I] R 1

To find the inverse of an invertible matrix A , find a sequence of elementary row operations that reduces A to the identity andthen perform this same sequence of operations on I n to obtain

A1

__________________________________________________________


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Example 19 :

Given matrix

1 2 1

2 3 1

3 1 2

A

=

. Find the product of EA.

Let E is obtained from3 3 12 R R R= .

Solution

1 3 3

1 0 0 1 0 02

0 1 0 0 1 0

0 0 1 2 0 1

R R R I E

+ = = =

1 0 0 1 2 1 1 2 1

0 1 0 2 3 1 2 3 1

2 0 1 3 1 2 1 3 0

EA

= =

Theorem ROW OPERATIONS BY MATRIX MULTIPLICATION

Let A an m x n matrix , and let an elementary row operation be performed on A to yielad matrix B. Let E obtained from

I m ( I n) by performing the same row operation as was performedon A . Then B =EA .


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By Theorem

1 3 3

1 2 1 1 2 12

2 3 1 = 2 3 13 1 2 1 3 0

R R R A B EA

+ =

= =

_____________________________________________________________________________

_

A method for Inverting Matrices

E k . . . E 2 E 1 A = I n A1 = E k . . . E 2 E 1 I n = E k . . . E 2 E 1

A1

TheoremEvery elementary matrix is invertible, and the inverse is alsoan elementary matrix

Theorem EQUIVALENT STATEMENTS

If A is an n x n matrix, then the following statements are equivalent,that is, all true or all false.

(a) A is invertible(b) Ax = 0 has only the trivial solution(c) The reduced row-echelon form of A is I n (d) A is a product of elementary matrices


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A = E 1 1 E 2 1 . . . E k 1 I n = E 1 1 E 2 1 . . . E k 1

(Elemetary matrices are invertible)

Consider the following :

1 0 1

0 1 1

1 1 0

A

=

1 3 3

2 3 3

1 0 1 1 0 0

0 1 1 0 1 0

1 1 0 0 0 1

1 0 1 1 0 0

0 1 1 0 1 0

0 1 1 1 0 1

1 0 1 1 0 00 1 1 0 1 0

0 0 2 1 1 1

R R R

R R R

+

+

3 3

1 0 1 1 0 01

0 1 1 0 1 0 20 0 11/ 2 1/ 2 1/ 2

R R


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3 1 1

3 2 2

1 0 1 1/ 2 1/ 2 1/ 2

0 1 0 1/ 2 1/ 2 1/ 2

0 0 1 1/ 2 1/ 2 1/ 2

R R R

R R R

+ +

Example 20 :

Let A be a 3 3 X matrix such that 3 2 1 3 E E E A I = where1 E is obtained from 3 I by performing the operation

1 3 1 R R R +

2 E is obtained from 3 I by performing the operation

2 1 22 R R R +

3 E is obtained from 3 I by performing the operation 3 2 R R

Find A and A1

using elementary matrices.(Note : E 3E2 E1 A = I 3 A = E 1 1 E2 1 E3 1

A1 = E 3E2 E1)

Solution

13 1 1 1 1

1 0 0 1 0 1 1 0 10 1 0 0 1 0 ; 0 1 0

0 0 1 0 0 1 0 0 1

I R R R E E = + = =


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11 2 2 2 2

1 0 0 1 0 0 1 0

0 1 0 2 2 1 0 ; 2 1 00 0 1 0 0 1 0 0

I R R R E E = + = =

13 2 3 3

1 0 0 1 0 0 1 0 0

0 1 0 0 0 1 ; 0 0 1

0 0 1 0 1 0 0 1 0

I R R E E = = =

1 1 11 2 3

1 0 1 1 0 0 1 0 00 1 0 2 1 0 0 0 1

0 0 1 0 0 1 0 1 0

1 0 1 1 0 0 1 1 0

2 1 0 0 0 1 2 0 10 0 1 0 1 0 0 1 0

A E E E = =

= =

13 2 1

1 0 0 1 0 0 1 0 1

0 0 1 2 1 0 0 1 00 1 0 0 0 1 0 0 1

1 0 0 1 0 1 1 0 10 0 1 0 1 0 0 0 12 1 0 0 0 1 2 1 2

A E E E = =

= =

Try this


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1 1 0

2 0 1

0 1 0

1 0 10 0 12 1 2

=

_ _ _

_ _ _

_ _ _

Row Operation to find A 1 [ A | I ] [ I | A1]

Example 21 :

Find the inverse of 1 2 34 5 6

3 1 2

A =

Solution 1 2 3 1 0 0 1 2 3 1 0 0

4 5 6 0 1 0 0 3 6 4 1 0

3 1 2 0 0 1 0 5 11 3 0 1

5 201 0 11 0 0 3 31 2 3

4 1 4 10 1 2 0 0 1 2 0

3 3 3 311 5 3 11 5 30 0 3 0 0 3


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5 201 0 1 16 7 13 3 1 0 0 3 3

4 1 26 110 1 2 0 0 1 0 23 33 3 0 0 1 51111 5 13 30 0 1 1

3 3

1

16 7 13 3 16 7 31

26 11 2 26 11 63 3 311 5 3511 13 3

A

= =

Try this

1

16 7 3 1 2 3 _ _ _ 1 26 11 6 4 5 6 _ _ _ 3

11 5 3 3 1 2 _ _ _

A A

= =

_____________________________________________________________________________ _

Showing That a Matrix Is Not Invertible

Example 22 :


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Show that A is not invertible

1 6 4

2 4 1

1 2 5

A

=

Solution

1 6 4 1 0 0

2 4 1 0 1 0

1 2 5 0 0 1

1 6 4 1 0 0

0 8 9 2 1 0

0 8 9 1 0 1

1 6 4 1 0 0

0 8 9 2 1 0

0 0 0 1 1 1

There is a row of zeros . Therefore A is not invertible.

A Consequence of Invertibility

Let A =

1 2 5

3 2 2

4 4 5

. If A is invertible , then

1. the homogeneous system has only the trivial solution .

2 5 0

3 2 2 0

4 4 5 0

x y z

x y z

x y z

+ = + =

+ =


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2. the nonhomogeneous system has exactly one solution.

2 5 1

3 2 2 2

4 4 5 1

x y z

x y z

x y z

+ = + =

+ =

3. the linear system is consistent (the linear system has a solution )

1

2

3

2 5

3 2 2

4 4 5

x y z b

x y z b

x y z b

+ = + =

+ =

__________________________________________________________

1.6 Further Results on Systems of Equations and

Invertibility

Basic Theorem

TheoremEvery system of linear equations has no solutions, or hasexactly one solution, or has infinitely many solutions.

Linear Systems by Matrix Inversion

Theorem


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If A is an invertible n x n matrix, then for each n x 1 matrix b ,the system of equations Ax =b has exactly one solution,namely , x = A 1b

Solution of a Linear System Using A 1

Write as A x = b. Find A 1 then A 1(A x) = A 1b x = A 1b

Example 23 :

Consider the system of linear equations

1

2 10

2

x y

x z

y

= + =

=.

Solve using matrix inversion. ( using A1 )

Solution

A =

1 1 0

2 0 1

0 1 0

and A1 =

1 0 10 0 1

2 1 2

(from Example 18 )

1 0 1 1 3

0 0 1 10 2

2 1 2 2 4

=

x = 3 , y = 2 , z =

4


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Solving Two Linear System at Once

Example 24 :

Solve the systems

2 2

2 5 1

3 7 2 1

x y z

x y z

x y z

+ = + + = + =

and

2 12 5 13 7 2 0

x y z

x y z

x y z

+ =+ + = + =

Solution

Linear system with a common coefficient matrix

1 2 1 2 1 1 2 1 2 1

2 5 1 1 1 0 9 1 5 3

3 7 2 1 0 0 1 1 5 3

1 2 1 2 1 1 2 1 2 1

0 9 1 5 3 0 9 1 5 30 0 10 50 30 0 0 1 5 3

1 2 0 3 2 1 2 0 3 2

0 9 0 0 0 0 1 0 0 0

0 0 1 5 3 0 0 1 5 3


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1 0 0 3 2

0 1 0 0 00 0 1 5 3

First system : x = 3 , y = 0 , z = 5

Second system : x = 2 , y = 0 , z = 3 __________________________________________________________

Properties of Invertible Matrices

AB = I and BA = I for A & B are square

matrices

_____________________________________________________________________________ _

TheoremLet A be a square matrix,

(a) If B is a square matrix satisfying BA = I , then B = A 1

(b) If B is a square matrix satisfying AB = I , then B = A 1

Theorem EQUIVALENT STATEMENTS

If A is an n x n matrix, then the following statements areequivalent, that is, all true or all false.

a) A is invertible b) A x = 0 has only the trivial solutionc) The reduced row-echelon form of A is I n d) A is expressible as a product of elementary

matrices.e) A x = b is consistent for every nx1 matrix bf) A x = b has exactly one solution for every n x 1

matrix b

TheoremLet A and B be a square matrices of the same size.If AB is invertible, then A and B must also be invertible.


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__________________________________________________________

Determining Consistency by Elimination

Example 25:

Find the conditions that the bs must satisfy the system to beconsistent . ( solution exists )

(a) 1 2 3 1

1 2 3 2

1 2 3 3

2 2 2

3 5

4 7 2

x x x b

x x x b

x x x b

+ =+ + =

=

(b)

1 2 3 1

1 2 3 2

1 2 3 3

2

2 3

3 7 4

x x x b

x x x b

x x x b

+ + = + =

+ =


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Solution

(a)

11

2 2 1

33 1

11 1 122 2 2

33 5 1 0 2 4

24 7 2 20 3 6

bb

b b b

b b b

+

1 1

2 1 2 1

3 1 3 1 2 1

1 1 11 1 12 2

2 3 2 30 1 2 0 1 2

4 42 2 2 3

0 1 2 0 0 03 3 4

b b

b b b b

b b b b b b

+ +

3 1 2 12 2 3Therefore 03 4

b b b b+ =

3 1 2 14 8 6 9

012 12

b b b b+ + =

1 2 36 4 0b b b + + =

1 236

4b b

b =


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(b)

1 1

2 1 2

3 3 1

1 1 2 1 1 21 2 3 0 1 5

3 7 4 0 10 2 3

b b

b b b

b b b

+

( ) ( )

1 1

1 2 1 2

3 1 1 2 3 1

1 1 2 1 1 2

0 1 5 0 1 5

0 10 2 3 0 0 52 10 3

b b

b b b b

b b b b b b

+

( ) ( )

1

1 2

1 2 3 1

1 1 2

0 1 5

0 0 1 10 3

52

b

b b

b b b b

+

Therefore b3 has no condition for the linear system to beconsistent.

1.7 Application to Electrical Networks

Objective: Be able to obtain systems of linear equations whosesolution yield the currents flowing in an electrical circuit basedon the basic laws of electrical circuits.

The electrical circuits consist of two basic components


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1. Electrical sources create currents in an electricalcircuit

2. Resistors limits the magnitudes of the currents

There are 3 basic quantities associated with electrical circuits:

- electrical potential (E) in volt, V- resistance (R) on ohms, - current (I) in amperes, A

Electrical potential is associated with two points in an electricalcircuit and is measured by a device called voltmeter .

In an electrical circuit, the electrical potential between two points is called the voltage drop.

The current and voltage drop can be positive (+ve) or negative( ve).

The flow of current is governed by:

1. ohms Law the voltage drop across a resistor is the product of the current passing through it and its

resistance E = IR

2. Kirchhoffs current Law The sum of the currentsflowing into any point equals the sum of the currentsflowing out from the point.


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3. Kirchhoffs voltage Law Around any closed loop, thealgebraic sum of the voltage drops is zero.

_________________________________________________ _

Example 1 :

Find the currents in the circuit.

Solution

2 1 3

2 1

2 3

3 2 14 03 4 24 0

I I I

I I

I I

= ++ =+ =

1 2 3

1 2

2 3

02 3 14

3 4 24

I I I

I I

I I

+ = + =

+ =

1 1 1 0

2 3 0 14

0 3 4 24

2 1 2

1 1 1 0

0 5 2 142

0 3 4 24 R R R

= +


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3 2 3

3 3

1 1 1 0

0 5 2 143 5

0 0 26 78

1 1 1 0

0 5 2 1410 0 1 326

R R R

R R

= +

=

3

2 3 2 2

1 2 3 1 1

3

205 2 14 5 2(3) 14 45

0 4 3 0 1

I

I I I I

I I I I I

=

= = = =

+ = + = =

1 2 31 , 4 , 3 I I I = = =

______________________________________________

Example 2 :

Consider the electrical network shown in Figure 1. ApplyKirchhoffs Law to find a system of linear equations relating

the currents I 1, I 2 and I 3 .Determine the currents in the circuit.


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Solutions :

1 2 3

1 2

2 3

02 4

4 22

I I I

I I

I I

+ = =

+ =

1

2

3

1 1 1 0

2 1 0 4

0 1 4 22

I

I

I

=

1 2 33 , 2 , 5 I I I = = =


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1.8 Diagonal and Triangular Matrices

Diagonal MatricesA diagonal matrix is a square n x n matrix whose nondiagonal entries are zero .

Example 1

1 0 0

0 5 0

0 0 2

D

=

____________________________________________________________________________________________________________________ _

Triangular MatricesA Triangular matrix is an m x n matrix whose entries either above or

below the main diagonal are zeros.

Example 2

1 2 3

0 5 6

0 0 2

A

=

1 0 0

4 5 0

3 1 2

B

=

Upper Triangular Lower Triangular

___________________________________________________________________________

Chapter_1-Slides(Sem 1 2013)

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