Chapter 2 … part1

Matrices

Linear Algebra

S 1

Ch2_2

2.1 Addition, Scalar Multiplication, and Multiplication of Matrices

DefinitionA matrix is a rectangular array of numbers .

The numbers in the array are called the elements of the matrix.Denoted by: A,B,… capital letter.

11 12 1

21 22 2

1 2

m

m

n n nm

a a a

a a aA

a a a

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2.1 Addition, Scalar Multiplication, and Multiplication of Matrices

• aij: the element of matrix A in row….. and column …… we say it is in the ………………..• The size of 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 column is called a ……….. matrix.• For a square nn matrix A, the main diagonal is: ……………….• We can denote the matrix by ……………..

Note:

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Definition

Two matrices are equal if:

1) …………………..

2) ……………………..

Example 1

2 5

3 0

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

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

3) .............. .

A

A size is

A is a matrix

is the main diagonal

3 1 7 1

1 2 2 4

0 0 3 1

' .........

B

it s size is

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Addition of Matrices

Definition• If A and B be matrices of the …………….. then the

sum A + B=C will be of the ……….. size and

……………………

• If

• Let A be a matrix and k be a scalar. The scalar multiple of A by k , denoted ………… will be the same size as A.

……………………

•The matrix (-1)A= -A called the …………… of A.

•Let A and B of the same size then: A - B= A +(-B)=C and:

……………………

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

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Example 2.72

45 and ,813652 ,320

741Let

CBA

Determine A + B , 3A , A + C , A-B

Solution

3

(3

(1)

(2)

)

(4)

A

A

A B

C

A B

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Definition• A ……. matrix all of it’s elements are zero. If the zero matrix is of a square size n×n it will be denoted by .

0ij

0n

Theorem2.2:Let A,B,C be matrices, be scalars.Assume that the size of the matrices are such that the operations can be performed, let 0 be the zero matrix.

Properties of matrix addition and scalar multiplication:

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

Solution

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Multiplication of MatricesDefinition1) If the number of ……….. in A = the number of …….. in B.

The product AB then exists. Let A: …….. matrix, B: ………. matrix,The product matrix C=AB is a ………. matrix.

2) If the number of ………. in A the number of …….. in B then The product AB ……………..

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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 bAB

a a a b b b

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

If AB C

then c

Note:

Example 4

Let C = AB, Determine c23.

105

237 and 4312 BA

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Example 5

3 1 2 7 2Let , Determine , .

4 1 5 6 3A B AB BA

Solution

Note. In general, ……………

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Definition1) A …….. matrix is a matrix in which all the elements are zeros.

2) A ……….. matrix is a square matrix in which all the elements ……………………………………...

3) An ……….. matrix is a diagonal matrix in which every element in the main diagonal is …….

matrix zero

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

Ch2_14

Theorem 2.1Let A be m n matrix and Omn be the zero m n matrix. Let B be an n n square matrix. On and In be the zero and identity n n matrices. Then:1) A + Omn = Omn + A = …….2) BOn = OnB = ………3) BIn = InB = ………Example 6

.4312 and 854

312Let

BA

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

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

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

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Let A, B, and C be matrices and k be a scalar. Assume that the size of the matrices are such that the operations can be performed.

Properties of Matrix Multiplication

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 =……… (where In is the identity matrix)

5. k(AB) = ………= ………

Note: AB BA in general. Multiplication of matrices is not commutative.

Theorem 2.2 -2

2.2 Algebraic Properties of Matrix Operations

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Example 7

.014

and ,201310 ,13

21Let

CBA Compute ABC.

Solution

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In algebra we know that the following cancellation laws apply.

If ab = ac and a 0 then ………..

If pq = 0 then ……….. or ……….

However the corresponding results are not true for matrices.

AB = AC ………………. that B = C.

PQ = O ………………… that P = O or Q = O.

Note:

Example 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

Ch2_18

Powers of Matrices

Theorem 2.3

If A is an n n square matrix and r and s are nonnegative integers, then

1. ArAs = ……….

2. (Ar)s = ……….

3. A0 = ……… (by definition)

Definition

If A is a square matrix and k is a positive integer, then

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

Ch2_19

Example 9

. compute ,01

21 If 4AA

Solution

Example 10 Simplify the following matrix expression.

ABBABABBAA 57)2(3)2( 22 Solution

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Idempotent and Nilpotent Matrices

Definition

A square matrix A is said to be:• ………………. if …………..• ………………. if there is a positive integer p s.t ……….…

The least integer p such that Ap=0 is called the ……………………. of the matrix.

3 6(1)

1 2A

Example 113 9

(2) 1 3

B

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2.3 Symmetric Matrices

Definition

The …………….. of a matrix A, denoted ………, is the matrix whose ………….. are the ………. of the given matrix A.

Example 12

.431 and ,654721 ,08

72

CBA

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

Determine the transpose of the following matrices:

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Theorem : Properties of Transpose

Let A and B be matrices and k be 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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Symmetric Matrix

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

DefinitionLet 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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Example 14

Proof

Let A and B be symmetric matrices of the same size.

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

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Example 15

Proof

Let A and B be symmetric matrices of the same size.

Prove that the product AB is symmetric if and only if AB = BA.

Ch2_26

Example 16

Proof

Let A be a symmetric matrix. Prove that A2 is symmetric.

Ch2_27

DefinitionLet A be a square matrix. The ………… of A denoted by …….. is the …………………………………. of A.

Let A and B be matrices and k be a scalar. Assume that the sizes of the matrices are such that the operations can be performed.

1. tr(A + B) = …………………..

2. tr(kA) = ………….

3. tr(AB) = …………

4. tr(At) = …………..

Theorem : Properties of Trace .