LinAlg-Chapter2 part1 s1.ppt

8/10/2019 LinAlg-Chapter2 part1 s1.ppt

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Chapter 2 part12.1 2.2 2.3

Matrices

Linear Algebra

S 1


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Ch2_2

2.1 Ad dit ion, Scalar Mu lt ip l icat ion, and Mult ip l icat ion o f Matr ices

Definition

Amatrix is a rectangular array of numbers.

The numbers in the array are called the elementsof the matrix.

letter.capitalDenoted by: A,B,

11 12 1

21 22 2

1 2

m

m

n n nm

a a a

a a aA

a a a


3/27

Ch2 3

2.1 Ad dit ion, Scalar Mu lt ip l icat ion, and Mult ip l icat ion o f Matr ices

aij: the element of matrix Ain row.. and column

we say it is in the ..

The sizeof a matrix = number of ... number of ... = .

If n=m the matrix is said to be a .. matrix with size = ...or .

A matrix that has one row is called a matrix. A matrix that has one columnis called a ..matrix.

For a square nnmatrixA, the main diagonalis: .

We can denote the matrix by ..

Note:


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Ch2_4

Definition

Two matrices are equalif:

1) ..

2) ..

Example 1

2 5

3 0

1) ................

2) ............... .3) .............. .

A

A size is

A is a matrixis the main diagonal

3 1 7 11 2 2 4

0 0 3 1

' .........

B

it s size is


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Ch2_5

Addition of Matrices

Definition

IfAandBbe matrices of the .. then the

sumA+B=C will be of the .. size and

If

LetAbe a matrix and k be a scalar. The scalar multipleofAby

k , denoted will be the same size asA.

The matrix (-1)A= -A called the of A.LetAandBof the same size then:A-B= A+(-B)=Cand:

..................A size B size A B


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Ch2_6

Example 2

.7245

and,813652

,320741

Let

CBA

DetermineA+B, 3A ,A+ C , A-B

Solution

3

(3

(1)

(2)

)

(4)

A

A

A B

C

A B


7/27Ch2_7

Definition

A. matrix all of its elements are zero. If the zero

matrixis of a square size nn it will be denoted by .

0ij0

n

Theorem2.2:

Let A,B,C bematrices, be scalars.

Assume that the size of the matrices are such that the operations

can be performed, let 0 be the zero matrix.

Propert ies of matr ix addit ion and s calar mu lt ip l icat ion :

1 2,k k

1) ............

2) ...............

3) 0 0 ........

A B

A B C

A A

1

1 2

1 2

4) .............

5) .............

6) ................

k A B

k k C

k k A


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2 3 5A B C

Example 3

Compute the linear combination: for:

1 3 3 7 0 2, ,

4 5 2 1 3 1A B C

Solut ion


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Mult ip l icat ion o f Matr ices

Definition

1) If the number of .. in A= the number of .. inB.

The productABthen exists.

LetA: ..matrix,B: .matrix,

The product matrix C=AB is a .matrix.

2) If the number of .in A the number of ..inB

then The product AB ..


10/27Ch2_10

11 12 1 11 12 1

21 22 2 21 22 2

1 2 1 2

11 12 1 11 12 1

21 22 2 21 22 2

1 2 1 2

,

m k

m k

n n nm m m mk

m k

m k

n n nm m m mk

a a a b b b

a a a b b bif A B

a a a b b b

a a a b b b

a a a b b bA B

a a a b b b


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..............................................ij

If AB C

then c

Note:

Examp le 4

Let C=AB, Determine c23.

105237

and4312

BA


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Example 5

3 1 2 7 2

Let , Determine , .4 1 5 6 3A B AB BA

Solut ion

Note.In general,


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Ch2_13

Definition

1) A ..matrixis a matrix in which all the elements are zeros.

2) A ..matrixis a square matrix in which all the

elements ...

3) An ..matrixis a diagonal matrix in which every

element in the main diagonal is .

matrixzero

000

000000

mn0

11

22

0 0

0 0

0 0

................. matrix A

nn

a

aA

a

1 0 0

0 1 0

0 0 1

............... matrix

nI

Special Matrices


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Ch2_14

Theorem 2.1

LetAbe mnmatrix and Omnbe the zero mnmatrix. LetBbe

an n

nsquare matrix. OnandInbe the zero and identity n

nmatrices. Then:

1) A+ Omn= Omn+A= .

2) BOn= OnB= 3) BIn=InB=

Examp le 6

.4312

and854312

Let

BA

23 ...............A O

2 .................BO

2 ...........BI


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Ch2_15

LetA,B, andCbe matrices and kbe a scalar. Assume that the size of

the matrices are such that the operations can be performed.

Properties of Matr ix Mul tipli cation

1.A(BC) = . Associative property of multiplication

2.A(B+ C) = Distributive property of multiplication

3. (A+B)C= Distributive property of multiplication

4.AIn=InA= (whereInis the identity matrix)

5. k(AB) = = Note:ABBA in general. Multiplication of matrices is not

commutative.

Theorem 2.2 -2

2.2 A lgebraic Propert ies of Matr ix Operat ions


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Ch2_16

Example 7

.

0

14

and,

201

310,

13

21Let

CBA ComputeABC.

Solut ion


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Ch2_17

In algebra we know that the following cancellation laws apply.

If ab= acand a 0 then ..Ifpq= 0 then ..or .

However the corresponding results are not true for matrices.

AB=AC.thatB= C.

PQ= Othat P= Oor Q= O.

Note:

Examp le 81 2 1 2 3 8

(1) Consider the matrices , , and .2 4 2 1 3 2

Observe that .................................., but ................

A B C

1 0 0 0(2) Consider the matrices , and .

0 0 0 1

Observe that ........................, but ........................

P Q


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Ch2_18

Powers of Matrices

Theorem 2.3

IfAis an nn square matrix andrandsare nonnegative

integers, then1. ArAs= .

2. (Ar)s= .

3. A0= (by definition)

Defini t ion

IfAis a squarematrix and kis a positive integer, then

...........................kA


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Ch2_19

Example 9

.compute,

01

21If 4AA

Solution

Example 10 Simplify the following matrix expression.

ABBABABBAA 57)2(3)2( 22

Solution


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Ch2_20

Idempo tent and Nilpotent Matr ices

Defini t ion

A square matrixAis said to be:

.if ..

.if there is a positive integerps.t .

The least integerpsuch thatAp=0 is called the

. of the matrix.

3 6

(1) 1 2A

Example 11

3 9(2)

1 3B


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Ch2_21

2.3 Symmetr ic Matr ices

Definition

The..of a matrixA, denoted , is the matrixwhose .. are the . of the given matrixA.

Example 12

.431and,654721

,0872

CBA

i.e., : : ..............tA m n A

Determine thetransposeof the following matrices:


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Ch2_22

Theorem :Propert ies of Transpose

LetAandBbe matrices and kbe a scalar. Assume that the sizes

of the matrices are such that the operations can be performed.

1. (A+B)t= ... Transpose of a sum

2. (kA)t= ... Transpose of a scalar multiple

3. (AB)t= ... Transpose of a product

4. (At)t= ...


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Ch2_23

Symmetr ic Matr ix

1 0 2 40 ....... 4

2 5 0 7 3 9

, 1 7 ....... ,5 4 2 3 2 3...... 8 3

4 9 3 6

match

match

Definition

Let A be a square matrix:

1) If ...then A called...matrix.

2) If ...then A called...matrix.

Example 13 symmetric matrices


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Ch2_24

Example 14

Proof

Let AandBbe symmetric matrices of the same size.

C = aA+bB, a,b are scalars. Provethat C is symmetric.


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Ch2_25

Example 15

Proof

Let AandBbe symmetric matrices of the same size.

Prove that the productABis symmetric if and only ifAB=BA.


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Ch2_26

Example 16

Proof

Let Abe a symmetric matrix. Prove thatA2is symmetric.


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Ch2 27

Definition

Let A be a square matrix. The of A denoted by ..isthe .of A.

LetAandBbe matrices and kbe a scalar. Assume that the sizesof the matrices are such that the operations can be performed.

1. tr(A + B) = ..

2. tr(kA) = .

3. tr(AB) =

4. tr(At) = ..

Theorem :Propert ies of Trace .

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LinAlg-Chapter2 part1 s1.ppt