Chapter 1

Chapter 1

Matrices and Systems of Equations

1 Systems of Linear Equations

)1(

2211

22222121

11212111

mnmnmm

nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

Where the aij’s and bi’s are all real numbers, xi’s are variables . We will refer to systems of the form (1) as m×n linear systems.

Definition Inconsistent : A linear system has no solution.Consistent : A linear system has at least one solution.

Example( ) xⅰ 1 + x2 = 2 x1 − x2 = 2

( ) ⅱ x1 + x2 = 2 x1 + x2 =1

( ) ⅲ x1 + x2 = 2 −x1 − x2 =-2

Definition Two systems of equations involving the same variables are said to be equivalent if they have the same solution set.

Three Operations that can be used on a system to obtain an equivalent system:

Ⅰ. The order in which any two equations are written may be interchanged.

Ⅱ. Both sides of an equation may be multiplied by the same nonzero real number.

Ⅲ. A multiple of one equation may be added to (or subtracted from) another.

n×n Systems

Definition

A system is said to be in strict triangular form if in the kthequation the coefficients of the first k-1 variables are all zero and the coefficient of xk is nonzero (k=1, …,n).

42

2

123

3

32

321

x

xx

xxx

is in strict triangular form.

Example The system

Example Solve the system

1

432

33

32

321

321

321

xxx

xxx

xxx

Elementary Row Operations:Ⅰ. Interchange two rows.Ⅱ. Multiply a row by a nonzero real number.Ⅲ. Replace a row by its sum with a multiple of another row.


3223

1242

6

0

4321

4321

4321

432

xxxx

xxxx

xxxx

xxx

2 Row Echelon Form

1

1

1

1

1

42211

31100

30022

10011

11111

0

1

3

0

1

31100

31100

52200

21100

11111

0

1

3

0

1

10000

10000

10000

21100

11111

3

4

3

0

1

00000

00000

10000

21100

11111

pivotal row

pivotal row

Definition

A matrix is said to be in row echelon form

ⅰ. If the first nonzero entry in each nonzero row is 1.

ⅱ. If row k does not consist entirely of zeros, the numb

er of leading zero entries in row k+1 is greater than the

number of leading zero entries in row k.

ⅲ. If there are rows whose entries are all zero, they are

below the rows having nonzero entries.

Example Determine whether the following matrices arein row echelon form or not.

010

000

000

100

321

400

530

642

100

310

241

01

10

0000

3100

0131

Definition

The process of using operations , , Ⅰ Ⅱ Ⅲ to transform a l

inear system into one whose augmented matrix is in ro

w echelon form is called Gaussian elimination.

Definition

A linear system is said to be overdetermined if there ar

e more equations than unknows.

A system of m linear equations in n unknows is said to

be underdetermined if there are fewer equations than u

nknows (m<n).

Example

232

322

2

)(

22

3

1

)(

54321

54321

54321

21

21

21

xxxxx

xxxxx

xxxxx

b

xx

xx

xx

a

Definition

A matrix is said to be in reduced row echelon form if:

ⅰ. The matrix is in row echelon form.

ⅱ. The first nonzero entry in each row is the only nonz

ero entry in its column.

Homogeneous SystemsA system of linear equations is said to be homogeneous if the constants on the right-hand side are all zero.

Theorem 1.2.1 An m×n homogeneous system of linear equations has a nontrivial solution if n>m.

3 Matrix Algebra

Matrix Notation

mnmm

n

n

aaa

aaa

aaa

A

21

22221

11211

VectorsVectors

row vector row vector

column vector column vector

nxxxX ,,, 21 1×n matrix matrix

nx

x

x

X2

1

n×1 matrix matrix

Definition

Two m×n matrices A and B are said to be equal if aij=bij f

or each i and j.

Scalar MultiplicationIf A is a matrix and k is a scalar, then kA is the matrix

formed by multiplying each of the entries of A by k.

Definition

If A is an m×n matrix and k is a scalar, then kA is the m×

n matrix whose (i, j) entry is kaij.

Matrix AdditionTwo matrices with the same dimensions can be added

by adding their corresponding entries.

Definition

If A=(aij) and B=(bij) are both m×n matrices,then the sum

A+B is the m×n matrix whose (i, j) entry is aij+bij for eac

h ordered pair (i, j).

1 1 2 1 0 02 0 3 , 0 1 0

1 1 2 0 0 1A I

2 3A I

Example

Let

Then calculate 。

Matrix Multiplication

Definition

If A=(aij) is an m×n matrix and B=(bij) is an n×r matrix, th

en the product AB=C=(cij) is the m×r matrix whose entri

es are defined by

ccijij = = aaii11bb11jj + + aaii22bb22jj +…+ +…+ aaiinnbbnnjj = = aaiikkbbkkjj. .

kk=1=1

nn

Example

,021114

,012301

BA then calculate AB.1. If

2. If ,1111,22

11

BA then calculate AB and BA.

Matrix Multiplication and Linear SystemsCase 1 One equation in Several Unknows

If we let and

then we define the product AX by

)( 21 naaaA

nx

x

x

X2

1

nnxaxaxaAX 2211

Case 2 M equations in N Unknows

If we let and

then we define the product AX by

mnmm

n

n

aaa

aaa

aaa

A

21

22221

11211

nx

x

x

X2

1

nmnmm

nn

nn

xaxaxa

xaxaxa

xaxaxa

AX

2211

2222121

1212111

Definition

If a1, a2, … , an are vectors in Rm and c1, c2, … , cn are scalars,

then a sum of the form

c1a1+c2a2+‥‥cnan

is said to be a linear combination of the vectors a1, a2, … , an .

Theorem 1.3.1 (Consistency Theorem for Linear

Systems)

A linear system AX=b is consistent if and only if b can be

written as a linear combination of the column vectors of

A.

Theorem 1.3.2 Each of the following statements is vali

d for any scalars k and l and for any matrices A, B and

C for which the indicated operations are defined.

1. A+B=B+A

2. (A+B)+C=A+(B+C)

3. (AB)C=A(BC)

4. A(B+C)=AB+AC

5. (A+B)+C=AC+BC

6. (kl)A=k(lA)

7. k(AB)=(kA)B=A(kB)

8. (k+l)A=kA+lA

9. k(A+B)=kA+kB

The Identity Matrix

Definition

The n×n identity is the matrix where)( ijI

jiif

jiifij 0

1

Matrix Inversion

Definition

An n×n matrix A is said to be nonsingular or invertible if

there exists a matrix B such that AB=BA=I.

Then matrix B is said to be a multiplicative inverse of A.

Definition

An n×n matrix is said to be singular if it does not have a

multiplicative inverse.

Theorem 1.3.3 If A and B are nonsingular n×n matrices,

then AB is also nonsingular and (AB)-1=B-1A-1

The Transpose of a Matrix

Definition

The transpose of an m×n matrix A is the n×m matrix B d

efined by

bji=aij

for j=1, …, n and i=1, …, m. The transpose of A is denote

d by AT.

Algebra Rules for Transpose:1. (AT)T=A

2. (kA)T=kAT

3. (A+B)T=AT+BT

4. (AB)T=BTAT

Definition

An n×n matrix A is said to be symmetric if AT=A.

4. Elementary Matrices

If we start with the identity matrix I and then perform

exactly one elementary row operation, the resulting matrix

is called an elementary matrix.

Type I. An elementary matrix of type I is a matrix obtained by interchanging two rows of I.

Example Let

100

001

010

1E and let A be a 3×3 matrix

then

333231

131211

232221

333231

232221

131211

1

100

001

010

aaa

aaa

aaa

aaa

aaa

aaa

AE

333132

232122

131112

333231

232221

131211

1

100

001

010

aaa

aaa

aaa

aaa

aaa

aaa

AE

Type II. An elementary matrix of type II is a matrix obtained by multiplying a row of I by a nonzero constant.

Example Let

300

010

001


then

333231

232221

131211

333231

232221

131211

2

333300

010

001

aaa

aaa

aaa

aaa

aaa

aaa

AE

333231

232221

131211

333231

232221

131211

2

3

3

3

300

010

001

aaa

aaa

aaa

aaa

aaa

aaa

AE

Type III. An elementary matrix of type III is a matrix obtained from I by adding a multiple of one row to another row.

Example Let

100

010

301


333231

232221

331332123111

333231

232221

131211

3

333

100

010

301

aaa

aaa

aaaaaa

aaa

aaa

aaa

AE

33313231

23212221

13111211

333231

232221

131211

3

3

3

3

100

010

301

aaaa

aaaa

aaaa

aaa

aaa

aaa

AE

In general, suppose that E is an n×n elementary matri

x. E is obtained by either a row operation or a column op

eration.

If A is an n×r matrix, premultiplying A by E has the

effect of performing that same row operation on A. If B

is an m×n matrix, postmultiplying B by E is equivalent

to performing that same column operation on B.

Example

Let ,

Find the elementary matrices ，， such that .1P 2P 1 2B PAP

11 12 13

21 22 23

31 32 33

a a aA a a a

a a a

31 32 33 33

21 22 23 23

11 12 13 13

333

a a a aB a a a a

a a a a

Theorem 1.4.1 If E is an elementary matrix, then E is

nonsingular and E-1 is an elementary matrix of the

same type.

Definition

A matrix B is row equivalent to A if there exists a finite

sequence E1, E2, … , Ek of elementary matrices such that

B=EkEk-1‥‥E1A

Theorem 1.4.2 (Equivalent Conditions for Nonsingularity)

Let A be an n×n matrix. The following are equivalent:

(a) A is nonsingular.

(b) Ax=0 has only the trivial solution 0.

(c) A is row equivalent to I.

Theorem 1.4.3 The system of n linear equations in n

unknowns Ax=b has a unique solution if and only if A

is nonsingular.

If A is nonsingular, then A is row equivalent to I and

hence there exist elementary matrices E1, …, Ek such

that

EkEk-1‥‥E1A=I multiplying both sides of this

equation on the right by A-1

EkEk-1‥‥E1I=A-1

Thus (A I) (I A-1) row operations

A method for finding the inverse of a matrix

Example Compute A-1 if

322

021

341

A


8322

122

1234

321

21

321

xxx

xx

xxx

Diagonal and Triangular Matrices

An n×n matrix A is said to be upper triangular if aij=0 for i>j and lower triangular if aij=0 for i<j.

An n×n matrix A is said to be diagonal if aij=0 whenever i≠j .

A is said to be triangular if it is either upper triangular or lower triangular.

5. Partitioned Matrices

C=

1 -2 4 1 3

2 1 1 1 1

3 3 2 -1 2

4 6 2 2 4

C11 C12

= C21 C22

-1 2 1

B= 2 3 1

1 4 1

=(b1, b2, b3)

AB=A(b1, b2, b3)=(Ab1, Ab2, Ab3)

In general, if A is an m×n matrix and B is an n×r that has

been partitioned into columns (b1, …, br), then the block

multiplication of A times B is given by

AB=(Ab1, Ab2, … , Abr)

If we partition A into rows, then

:),(

:),2(

:),1(

ma

a

a

A

Then the product AB can be partitioned into rows as follows:

Bma

Ba

Ba

AB

:),(

:),2(

:),1(

Block Multiplication

Let A be an m×n matrix and B an n×r matrix.

Case 1 B=(B1 B2), where B1 is an n×t matrix and B2

is an n×(r-t) matrix.

AB= A(b1, … , bt, bt+1, … , br) = (Ab1, … , Abt, Abt+1, … , Abr) = (A(b1, … , bt), A(bt+1, … , br)) = (AB1 AB2)

Case 2 A= ,where A1 is a k×n matrix and A2

is an (m-k)×n matrix.

2

1

A

A

Thus

BA

BAB

A

A

2

1

2

1

Case 3 A=(A1 A2) and B= , where A1 is an m×s matrix

and A2 is an m×(n-s) matrix, B1 is an s×r matrix and B2 is an

(n-s)×r matrix.

2

1

B

B

Thus 22112

121 BABAB

BAA

Case 4 Let A and B both be partitioned as follows ：

A11 A12 kA= A21 A22 m-k

s n-s

B11 B12 sB= B21 B22 n-s

t r-t

2222122121221121

2212121121121111

2221

1211

2221

1211

BABABABA

BABABABA

BB

BB

AA

AA

Then

In general, if the blocks have the proper

dimensions, the block multiplication can be

carried out in the same manner as ordinary

matrix multiplication.

,

000100311132

401

,

040006110023010

52001

BA

Example

Let

Then calculate AB.

Example

Let A be an n×n matrix of the form

22

11

0

0

A

A

where A11 is a k×k matrix (k<n). Show that

A is nonsingular if and only if A11 and A22

are nonsingular.

Documents

Chapter 1