linear algebra(ppt)updated

8/3/2019 Linear Algebra(Ppt)Updated

1/112

Chapter 1

Systems of Linear Equations

and Matrices


2/112

1.1) System of Linear Equations

A system of m linear equations in n

variables is a collection of equations of

the form

This is also referred as an mxn linear

system

i1


3/112

Slide 2

i1 idaqqa, 2/14/2010


4/112

A solution to a system of equations with n

variables is an ordered sequence

such that each equation is satisfied for

. The general solution or

solution set is the set of all possible solutions.

),...,,( 21 nsss

nnsxsxsx !!! ,....,, 2211


5/112

The Elimination method

This method is also called Gaussian

elimination. To introduce this method we

need to define the triangular form of a linear

system.

An mxn linear system is triangular form

provided that the coefficients

for example

jiwheneveraij "! 0

!

!

!

2

53

12

3

32

321

x

xx

xxx


6/112

When a linear system is in triangular form ,

then the solution set can be obtained using a

technique called back substitution.

Example: for the linear system above we see

that

Def 2: Two linear systems are equivalent if they

have the same solutions.

1,11,2 123 !!! xxx


7/112

Example:

are equivalent.

!!

!

!!

!

253

12

122332

12

3

32

321

321

321

321

x

xx

xxx

andxxxxxx

xxx


8/112

Theorem 1)

Performing any one of the followingoperations on the linear system produces anequivalent linear system

1) interchanging any two equations.

2) multiplying any equation by a nonzero

constant 3) adding a multiple of one equation to

another.


9/112

A system of two equations

Two Equations Containing Two

Variables

The solution will be one of three cases:1. Exactly one solution, an ordered pair (x, y)

2. A dependent system with infinitely many solutions

3. No solution

The first two cases are called consistentsince there are solutions. The last case iscalled inconsistent


10/112

With two equations and two variables, the

graphs are lines and the solution (if there is one)

is where the lines intersect. Lets look at

different possibilities.consistent

inconsistent Dependent

consistent

Case 3: independent

The lines are

parallel so there

is no solution.

Case 1: The lines

intersect at a point

(x, y),

the solution

Case 2: The lines

coincide and there

are infinitely many

solutions (all points

on the line).


11/112

Example 1 page 6

Example 2 page 6

Three Equations Containing Three Variables


12/112

The solution will be one of three cases:

1. Exactly one solution, an ordered triple (x, y, z)

2. A dependent system with infinitely many solutions

3. No solution

With two equations and two variables, the

graphs were lines and the solution (if there was

one) was where the lines intersected. Graphs of

three variable equations are planes. Lets look

at different possibilities. Remember the solution

would be where all three planes all intersect.


13/112

Planesintersect at a

point:

consistent with

one solution


14/112

Planes intersect in a line: consistent

system called dependent with an infinite

number of solutions


15/112

Three parallel planes: no intersection so systemcalled inconsistent with no solution

Example 3 page 8 Example 4 page 9

Example 5 page 10, Example 6 page 11


16/112

HW Problem Page 12, page 13

3,11,21,23,27,31,35


17/112

1.2) Matrices and elementary row

operations

An mxn matrix is an array of numbers with m

rows and n columns.

For example:

is a 3X4 matrix

-

3142

20134232


18/112

Augmented Matrices

mnmnmm

nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

!

!

!

...

...

...

2211

22222121

11212111

////

-

mmnmm

n

n

baaa

baaa

baaa

...

...

...

21

222221

111211

////

The location of the +s, the

xs, and the =s can be

abbreviated by writing only

the rectangular array ofnumbers.

This is called the augmented

matrix for the system.

Note: must be written in thesame order in each equation

as the unknowns and the

constants must be on the

right.

1th column

1th row


19/112

Elementary Row Operations

The basic method for solving a system of linear equations is to replace

the given system by a new system that has the same solution set but

which is easier to solve.

Since the rows of an augmented matrix correspond to the equations

in the associated system. new systems is generally obtained in a series

of steps by applying the following three types of operations to

eliminate unknowns systematically. These are called elementary row

operations.1. Multiply an equation through by a nonzero constant.

2. Interchange two equation.

3. Add a multiple of one equation to another.


20/112

Example 3

Using Elementary row Operations(1/4)

0563

7172

92

!

!

!

zyx

zy

zyx

psecondtheto equationfirstthe

times2-add

0563

1342

92

!

!

!

zyx

zyx

zyx

-

0563

1342

9211

-

0563

17720

9211

pthirdtheto equationfirstthe

times3-add

pthirdthetorowfirstthetimes3-add

psecondthetorowfirstthetimes2-add


21/112

Example 3


0113

92

217

27

!

!

!

zy

zy

zyx

p

21byequation

secondthemultiply

27113

1772

92

!

!

!

zy

zy

zyx

-

271130

17720

9211

-

271130

10

9211

217

27

pthirdtheto equationsecondthe

times3-add

p thirdthetorowsecondthe

times3-add

p

2

1byrow

secondemultily th


22/112

Example 3


3

92

217

27

!

!

!

z

zy

zyx

p2-byequationthirdtheMultiply

23

21

217

27

92

!

!

!

z

zy

zyx

-

23

21

217

27

00

10

9211

-

3100

10

9211

217

27

pfirsttheto equationsecond

thetimes1-Add

p firstthetorowsecond

thetimes1-Add

p 2-byrow

thirdeMultily th


23/112

Example 3


32

1

!

!

!

z

y

x

psecondtheto equationthirdthe

timesandfirstthetoequationthirdthe

times-Add

27

211

32

17

2

7

235

211

!

!

!

z

zy

zx

-

3100

10

01

217

27

235

211

-

3100

2010

1001

p

secondthetorowthirdthetimes

andfirstthetorowthirdthetimes-Add

27

211

The solution x=1,y=2,z=3 is now evident.


24/112

Echelon Forms The matrix which have following properties is in reduced row-echelon

form (see Example 1, 2).

1. If a row does not consist entirely of zeros, then the first nonzero

number in the row is a 1. We call this a leader 1 (pivot).2. If there are any rows that consist entirely of zeros, then they are

grouped together at the bottom of the matrix.

3. In any two successive rows that do not consist entirely of zeros, theleader 1 in the lower row occurs farther to the right than the leader 1in the higher row.

4. Each column that contains a leader 1 has zeros everywhere else.

A matrix that has the first three properties is said to be in row-echelonform (Example 1, 2).

A matrix in reduced row-echelon form is of necessity in row-echelonform, but not conversely.


25/112

Example 1

Row-Echelon & Reduced Row-Echelon form

reduced row-echelon form:

-

-

-

-

00

00,

00000

00000

31000

10210

,

100

010

001

,

1100

7010

4001

row-echelon form:

-

-

-

10000

01100

06210

,

000

010

011

,

5100

2610

7341


26/112

Example 2

More on Row-Echelon and Reduced Row-Echelon form All matrices of the following types are in row-echelon form

( any real numbers substituted for the *s. ) :

-

-

-

-

*100000000

*0**100000

*0**010000

*0**001000

*0**000*10

,

0000

0000

**10

**01

,

0000

*100

*010

*001

,

1000

0100

0010

0001

-

-

-

-

*100000000

****100000

*****10000

******1000

********10

,

0000

0000

**10

***1

,

0000

*100

**10

***1

,

1000

*100

**10

***1

All matrices of the following types are in reduced row-echelon

form ( any real numbers substituted for the *s. ) :


27/112

Example

Solutions of 3 Linear Systems (a)

-

4100

2010

5001

(a)

4

2-

5

!

!

!

z

y

x

Solution (a)

the corresponding system of

equations is :

Suppose that the augmented matrix for a system of linear

equations have been reduced by row operations to the given

reduced row-echelon form. Solve the system.


28/112

Example 3

Solutions of 3 Linear Systems (b)

-

1000

0210

0001

(b)

Solution (b):

the last equation in the corresponding system of

equation is

Since this equation cannot be satisfied, there is no

solution to the system.

1000 321 ! xxx


29/112

Elimination Methods (1/7) We shall give a step-by-step elimination

procedure that can be used to reduce any matrix

to reduced row-echelon form.

-

156542

281261042

1270200


30/112

Elimination Methods (2/7) Step1. Locate the leftmost column that does not consist

entirely of zeros.

Step2. Interchange the top row with another row, to bring a

nonzero entry to top of the column found in Step1.

-

156542281261042

1270200

Leftmost nonzero column

-

156542

1270200

281261042The 1th and 2th rows in the preceding

matrix were interchanged.


31/112

Elimination Methods (3/7) Step3. If the entry that is now at the top of the column found in

Step1 is a, multiply the first row by 1/a in order to introduce a

leading 1.

Step4. Add suitable multiples of the top row to the rows below

so that all entires below the leading 1 become zeros.

-

156542

1270200

1463521

The 1st row of the preceding matrix

was multiplied by 1/2.

-

29170500

1270200

1463521-2 times the 1st row of the preceding

matrix was added to the 3rd row.


32/112

Elimination Methods (4/7) Step5. Now cover the top row in the matrix and begin again

with Step1 applied to the submatrix that remains. Continue

in this way until the entire matrix is in row-echelon form.

-

29170500

1270200

1463521

The 1st row in the submatrix was

multiplied by -1/2 to introduce a

leading 1.

-

29170500

60100

1463521

2

7

Leftmost nonzero column in

the submatrix


33/112

Elimination Methods (5/7) Step5 (cont.)

-

210000

60100

1463521

2

7

-5 times the 1st row of the submatrix was

added to the 2nd row of the submatrix to

introduce a zero below the leading 1.

-

10000

60100

1463521

2

1

2

7

-

10000

60100

1463521

2

1

2

7

The top row in the submatrix was covered,

and we returned again Step1.

The first (and only) row in the new

submetrix was multiplied by 2 to

introduce a leading 1.

Leftmost nonzero column in the

new submatrix

The entire matrix is now in row-echelon form.


34/112

Elimination Methods (6/7)

Step6. Beginning with las nonzero row and working upward, add

suitable multiples of each row to the rows above to introduce zeros

above the leading 1s.

-

210000

100100

703021

7/2 times the 3rd row of the preceding

matrix was added to the 2nd row.

-

210000

100100

1463521

-

210000

100100

203521-6 times the 3rd row was added to the

1st row.

The last matrix is in reduced row-echelon form.

5 times the 2nd row was added to the

1st row.


35/112

Elimination Methods (7/7)

Step1~Step5: the above procedure produces a row-

echelon form and is called Gaussian elimination.

Step1~Step6: the above procedure produces a reduced

row-echelon form and is called Gaussian-Jordan

elimination.

Every matrix has a unique reduced row-echelon form

but a row-echelon form of a given matrix is not unique.


36/112

Example 4 page 21

Example 5 page 22

Example 6 page 22

HW page 24-26:

7,11,13,18,19,25,27,35,41,45,49


37/112

1.3) Matrix Algebra

Def: A matrix is a rectangular array of numbers.

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


38/112

Examples of matrices

Some examples of matrices

Size

3 x 2, 1 x 4, 3 x 3, 2 x 1, 1 x 1

? A ? A4,31,

000

10

2

,3-012,

41

03

21

21

-

-

T

-

N

# rows

# columns

row matrix or row vector column matrix or

column vector

entries


39/112

Matrices Notation and Terminology(1/2)

A general m x n matrix A as

The entry that occurs in row i and column j of matrix A will

be denoted . If is real number, it is

common to be referred as scalars.

-

!

mnmm

n

n

aaa

aaa

aaa

A

...

...

...

21

22221

11211

///

ijij Aa or ija


40/112

A matrix A with n rows and n columns is called a square

matrix of order n, and the shaded entries

are said to be on the main diagonal of A.

Matrices Notation and Terminology(2/2)

-

nnnn

n

n

aaa

aaa

aaa

...

...

...

21

22221

11211

///

nnaaa ,,, 2211 .


41/112

Definition

Two matrices are defined to be equal if

they have the same size and their

corresponding entries are equal.

? A ? A

j.andiallforifonlyandifBAthen

size,samethehaveandIf

ijij

ijij

ba

bBaA

!!

!!


42/112

Example 2

Equality of Matrices

Consider the matrices

If x=5, then A=B.

For all other values of x, the matrices A and B are not equal.

There is no value of x for which A=C since A and C have different

sizes.

-

!

-

!

-

!

043

012,

53

12,

3

12CB

x

A


43/112

Operations on Matrices

If A and B are matrices of the same size, then the sum A+B

is the matrix obtained by adding the entries of B to the

corresponding entries of A.

Vice versa, the difference A-B is the matrix obtained by

subtracting the entries of B from the corresponding

entries of A.

Note: Matrices of different sizes cannot be added or

subtracted.

ijijijijij

ijijijijij

baBABA

baBABA

!!

!!

)()(

)()(


44/112

Example 3

Addition and Subtraction

Consider the matrices

Then

The expressions A+C, B+C, A-C, and B-C are undefined.

-

!

-

!

-

!

22

11,

5423

1022

1534

,

0724

4201

3012

CBA

-

!

-

!

51141

5223

2526

,

5307

3221

4542

BABA


45/112

Definition

If A is any matrix and c is any scalar, then the product

cA is the matrix obtained by multiplying each entry of

the matrix A by c. The matrix cA is said to be the

scalar multiple of A.

? A

ijijijij

caAccAa

!!

!

then,Aifnotation,matrixIn


46/112

Example 4

Scalar Multiples (1/2) For the matrices

We have

It common practice to denote (-1)B byB.

-

!

-

!

-

!

120

3

369,

531

720,

131

432CBA

-

!

-

!

-

!

40

1

123,

531

7201-,

262

8642

31CBA


47/112

Example 4

Scalar Multiples (2/2)


48/112

Theorem 4) properties of matrix addition and

scalar multiplication

Let A,B, and C be mxn matrices and cand d be

real numbers.

1) A+B=B+A

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

3) c(A+B)=cA+cB

4) (c+d)A=cA+dA 5) c(dA)=(cd)A


49/112

Theorem 4) Continue

A matrix, all of whose entries are zero, such as

is called a zero matrix .

6) A zero matrix will be denoted by , and A+0=0+A=A 7) For any matrix A, the matrix A, whose components are

negative of each component of A, is such that A+(-A)=-A+A=0

nmv0


50/112

Definition

If A is an mr matrix and B is an rn matrix, then theproduct AB is the mn matrix whose entries aredetermined as follows.

To find the entry in row i and column j of AB, singleout row i from the matrix A and column j from thematrix B .Multiply the corresponding entries fromthe row and column together and then add up theresulting products.


51/112

Example 5

Multiplying Matrices (1/2) Consider the matrices

Solution

Since A is a 2 3 matrix and B is a 3 4 matrix, the

product AB is a 2 4 matrix. And:


52/112

Example 5

Multiplying Matrices (2/2)


53/112

Examples 6

DeterminingW

hether a Product Is Defined Suppose that A ,B ,and C are matrices with the following sizes:A B C

3 4 4 7 7 3

Solution:

Then by (3), AB is defined and is a 3 7 matrix; BC is defined and is a 4

3 matrix; and CA is defined and is a 7 4 matrix. The products

AC ,CB ,and BA are all undefined.


54/112

Properties of Matrix Operations

For real numbers a and b ,we always have ab=ba, which is

called the commutative law for multiplication. For matrices,

however, AB and BA need not be equal.

Equality can fail to hold for three reasons:

The product AB is defined but BA is undefined.

AB and BA are both defined but have different sizes.

it is possible to have AB=BA even if both AB and BA are defined

and have the same size.


55/112

Example1

AB and BA Need Not Be Equal


56/112

Example2

Associativity of Matrix Multiplication


57/112

Properties of Zero Matrices Assuming that the sizes of the matrices are such

that the indicated operations can be

performed,the following rules of matrix

arithmetic are valid.


58/112

Def 4: If A is any mn matrix, then the transposeof A ,denoted by ,is defined to be the nm

matrix that results from interchanging the rows

and columns of A ; that is, the first column of

is the first row of A ,the second column of isthe second row of A ,and so forth.

Symmetric matrix: An nxn matrix is symmetric

provided that =

Example 5 page 36

T

A

TA

TA

TA A


59/112

Example

Some Transposes (1/2)


60/112

Example

Some Transposes (2/2)

Observe that

In the special case where A is a square matrix, thetranspose of A can be obtained by interchanging entriesthat are symmetrically positioned about the main diagonal.

See theorem 6 page 36.


61/112

Definition

If A is a square matrix, then the trace of

A ,denoted by tr(A), is defined to be the

sum of the entries on the main diagonal ofA .The trace of A is undefined if A is not a

square matrix.


62/112

Example 11

Trace of Matrix

HWPage 3739 : 16, 7,9,13,17,21,27,31,35,39


63/112

1.4) The Inverse of a square Matrix

Identity MatricesOf special interest are square matrices with 1s on the main

diagonal and 0s off the main diagonal, such as

A matrix of this form is called an identity matrix and is denotedby I .If it is important to emphasize the size, we shall write

for the nn identity matrix. If A is an mn matrix, then

A = A and A = A

Recall : the number 1 plays in the numerical relationships

a1 = 1a = a .

nI mI

nI


64/112

Example4

Multiplication by an Identity Matrix

mIRecall : A = A and A =A , as A is an mn matrix

nI


65/112

If R is the reduced row-echelon form of an nn

matrixA, then eitherR has a row of zeros orR is theidentity matrix .nI


66/112

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 said to be invertible

and B is called an inverse of A . If no such matrix B can be

found, then A is said to be singular.

Notation:1

!AB


67/112

Example5

Verifying the Inverse requirements


68/112

Example6

A Matrix with no Inverse


69/112

Properties of Inverses

It is reasonable to ask whether an invertible

matrix can have more than one inverse. The

next theorem shows that the answer is no

an invertible matrix has exactly one inverse .


70/112

Theorem 8


71/112

Theorem 9

IfA andB are invertible matrices of

the same size ,then AB is invertible

and

The result can be extended :

111 ! ABAB

l


72/112

Example

Inverse of a Product


73/112

Definition


74/112

Laws of Exponents

If A is a square matrix andr

ands are integers ,then

rssrsrsr AAAAA !! ,

E l


75/112

Example

Powers of a Matrix


76/112

Polynomial Expressions Involving Matrices

If A is a square matrix, say mm, and if

is any polynomial, then w define

where I is the mm identity matrix.

In words, p(A) is the mm matrix that resultswhen A is substituted for x in (1) and

is replaced by .

(1)10n

nxaxaaxp ! .

nnAaAaIaAp ! .10

0a Ia

0


77/112

Properties of the Transpose If the sizes of the matrices are such that the

stated operations can be performed,then

Part (d) of this theorem can be extended :


78/112

Invertibility of a Transpose

IfA is an invertible matrix,then is also invertible and

TT AA 11 !

TA


79/112

Example


80/112

1.5) Matrix Equations

mnmnmm

nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

!

!

!

...

...

...

2211

22222121

11212111

////

-

!

-

mnmnmm

nn

nn

b

b

b

xaxaxa

xaxaxa

xaxaxa

////

2

1

2211

2222121

1212111

...

...

...

-

!

-

-

mmmnmm

n

n

b

b

b

x

x

x

aaa

aaa

aaa

/////

2

1

2

1

21

22221

11211

...

...

...

Consider any system of m

linear equations in n unknowns.

Since two matrices are equal if

and only if their corresponding

entries are equal.

The m1 matrix on the left side

of this equation can be written

as a product to give:


81/112

1.5)Matrix Equations

If we designate these matrices by A ,x ,and b ,respectively,

the original system of m equations in n unknowns has been

replaced by the single matrix equation

The matrix A in this equation is called the coefficient matrixof the system. The augmented matrix for the system is

obtained by adjoining b to A as the last column; thus the

augmented matrix is


82/112

The determinant of a square matrix is a number

obtained in a specific manner from the matrix.

For a 1x1 matrix:

For a 2x2 matrix:

1.6) DeterminantsWhat is a determinant?

? A 1111 aAaA !! )det(;

211222112221

1211aaaaA

aaaaA !

-! )det(;

Product along red arrow minus product along blue arrow

E l 1


83/112

Example 1

7531A

-!

8537175

31)A !vv!!det(

Consider the matrix

Notice (1) A matrix is an array of numbers

(2) A matrix is enclosed by square brackets

Notice (1) The determinant of a matrix is a number

(2) The symbol for the determinant of a matrix is a pair of parallel

lines

Computation of larger matrices is more difficult

D li t l th d f 3 3 t i


84/112

ForONLY a 3x3 matrix write down the first two

columns after the third column

Duplicate column method for 3x3 matrix

3231

2221

1211

333231

232221

131211

aa

aa

aa

aaa

aaa

aaa

-

Sum of products along red arrow

minus sum of products along blue arrow

This technique works only for 3x3 matrices

332112322311312213

322113312312332211

aaaaaaaaa

aaaaaaaaa)A

!det(

aaa

aaa

aaa

A

333231

232221

131211

-

!

E l


85/112

Example

-

!

2

01A

21-

4

3-42

12

01

42

212

401

342

-

0 32 30 -8 8

Sum of red terms = 0 + 32 + 3 = 35

Sum of blue terms = 0 8 + 8 = 0

Determinant of matrix A= det(A) = 35 0 = 35

Finding determinant using inspection


86/112

Finding determinant using inspection

Special case. Iftwo rows ortwo columns are proportional (i.e. multiples of

each other), then the determinant of the matrix is zero

0

872423

872

!

because rows 1 and 3 are proportional to each other

If the determinant of a matrix is zero, it is called a singular matrix

C f t th d What is a cofactor?


87/112

If A is a square matrix

Cofactor method

The minor, Mij, of entry aij is the determinant of the submatrix that remains after the

ith row and jth column are deleted from A.

The cofactor of entry aij is Cij=(-1)(i+j) Mij

31233321

3331

2321

12 aaaaaa

aaM !!

3331

2321

1212aa

aaMC !!

aaa

aaa

aaa

A

333231

232221

131211

-

!

What is a cofactor?

What is a cofactor?


88/112

Sign of cofactor

What is a cofactor?

-

---

-

Find the minor and cofactor of a33

414020142M33 !vv!!

Minor

4MM)1(C 3333)33(

33 !!!Cofactor

-

!

2

01A

21-

4

3-42

Cofactor method of obtaining the


89/112

Cofactor method of obtaining thedeterminant of a matrix

The determinant of a n x n matrix A can be computed by multiplying ALL theentries in ANY row (or column) by theircofactors and adding the resulting

products. That is, for each andni1 ee nj1 ee

njnj2j2j1j1j CaCaCaA ! .)det(

Cofactor expansion along the ith

row

inini2i2i1i1 CaCaCaA ! .)det(

Cofactor expansion along the jth column

Example: evaluate det(A) for:


90/112

Example: evaluate det(A) for:

1

3

-1

0

0

4

5

1

2

0

2

1

-3

1

-2

3

A=

det(A)=(1)

4 0 1

5 2 -2

1 1 3

- (0)

3 0 1

-1 2 -2

0 1 3

+ 2

3 4 1

-1 5 -2

0 1 3

- (-3)

3 4 0

-1 5 2

0 1 1

= (1)(35)-0+(2)(62)-(-3)(13)=198

det(A) = a11C11 +a12C12 + a13C13 +a14C14

Example : evaluate


91/112

By a cofactor along the third column

Example : evaluate

det(A)=a13C13 +a23C23+a33C33

det(A)=

1 5

1 03 -1

-3

22

= det(A)= -3(-1-0)+2(-1)5(-1-15)+2(0-5)=25

det(A)=1 0

3 -1

-3* (-1)41 5

3 -1+2*(-1)5

1 5

1 0

+2*(-1)6


92/112

LetA be a square matrix

(a) ifA has a row of zeros or a column of zeros, then det(A)

= 0.

(b) det(A) = det(AT)


93/112

Triangular Matrix

IfA is an nxn triangular matrix (upper triangular,

lower triangular, or diagonal), then det(A) is the

product of the entries on the main diagonal of the

matrix ; that is, det(A) = a11a22an.

Example


94/112

Example

Determinant of an Upper Triangular

1296)4)(9)(6)(3)(2(

40000

89000

67600

1573-0

383-72

!!


95/112

Elementary Row Operations

LetA be an nxn matrix

(a) IfB is the matrix that results when a single row or singlecolumn ofA is multiplied by a scalar k, than det(B) = kdet(A)

(b) IfB is the matrix that results when two rows or twocolumns ofA are interchanged, then det(B) = - det(A)

(c) IfB is the matrix that results when a multiple of onerow ofA is added to another row or when a multiplecolumn is added to another column, then det(B) = det(A)


96/112

Example

Relationship Operation

det(B) =kdet(A)

The firstrow ofA ismultiplied by k.

det(B) =-det(A)

The first and secondrows are interchanged

333231

232221

131211

333231

232221

131211

kk

aaa

aaa

aaa

k

aaa

aaa

aaka

!

333231

232221

131211

333231

131211

232221

aaa

aaa

aaa

aaa

aaa

aaa

!

Example


97/112

Example

Relationship Operation

det(B) = det(A)

A multiple of

the secondrows ofA is

added to thefirstrow

333231

232221

131211

333231

232221

231322122111

aka

aaa

aaa

aaa

aaa

aaa

kaakaa

!


98/112

Matrices with Proportional Rows or

Columns IfA is a square matrix with two proportional rows ortwo proportional column, then det(A) = 0.


99/112

Example

Introducing Zero Rows The following computation illustrates the introduction of a row of

zeros when there are two proportional rows.

Each of the following matrices has two proportional rows or

columns; thus, each has a determinant of zero.

.

0

8411

5193

0

0

0

0

42-31

8411

519384-62

42-31

!!

.

1512-39-

4185

252-65-41-3

,

411

584-

72-1

,82-

41-

The second row is 2 times the first,so we added -2 times the first row to

the second to introduce a row of

zeros

E l


100/112

Example

Using Row Reduction to Evaluate a

Determinant(2/2)

.

165)1)(55)(3(

100

51032-1

)55)(3(

55-00

510

32-1

3

5-100

510

32-1

3

!!

!

!

!

-2 times the first row was

added to the third row.

-10 times the second row

was added to the third

row

A common factor of -55

from the last row was

taken through the

determinant sign.

E l


101/112

Example

Using Column Operation to Evaluate a

Determinant Compute the determinant of

Solution:

We can putA in lower triangular form in one step by adding -3

times the first column to the fourth to obtain

-

!

5-137

036

0

6072

3001

A

546)26)(7)(1(

26-137

0360

0072

0001

det)det( !!

-

!

A


102/112

Basic Properties of determinant(1/2)

Suppose thatA andB are nxn matrices and kis any scalar. We begin

by considering possible relationships between det(A), det(B), and

det(kA), det(A+B), and det(AB)

Since a common factor of any row of a matrix can be moved through

the det sign, and since each of the n row in kA has a common factor

ofk, we obtain

(1)

For example,

)det()det( AkkA n!

333231

232221

131211

3

333231

232221

131211

aaa

aaa

aaa

k

kakaka

kakaka

kakaka

!

( / )


103/112

Basic Properties of determinant(2/2)

There is no simple relationship exists between det(A),

det(B), and det(A+B) in general. In particular, we

emphasize that det(A+B) is usually notequal to

det(A)+det(B).


104/112

Example

Consider

]det[]det[]det[ BABA {

)det()det()det(

thus23;B)det(Aand8,det(B)1,det(A)haveWe

83

34

BA,31

13

,52

21

BABA

BA

{

!!!

-

!

-

!

-

!


105/112

LetA,B, and Cbe nxn matrices that differ only in asingle row, say the rth, and assume that the rth rowofCcan be obtained by adding correspondingentrues in the rth rows ofA andB. Then

det(C) = det(A)+det(B)

The same result holds for columns


106/112

Example

By evaluating the determinants

-

-

!

-

110

302

571

det

741

302

571

det

)1(71401

302

571

det


107/112

Example

Determinant Test for Invertibility Since the first and third rows of

are proportional, det(A) = 0. Thus,A isnot invertible

-

!

642

101

321

A


108/112

IfA andB are square matrices of the same size, then

det(AB)=det(A)det(B)


109/112

Corollary 1

IfA is invertible, then

)det(1)det( 1

AA !


110/112

Cramers Rule

If is a system of n linear equations in n unknowns such

that , then the system has a unique solution. This solution is

Where is the matrix obtained by replacing the entries in the column of

A by the entries in the matrix

bx !A0)det( {A

)det(

)det(

,)det(

)det(

,)det(

)det( 33

2

2

1

1 A

A

xA

A

xA

A

x !!!

jA

-

!

nb

b

b

/

2

1

b

Example


111/112

Example

Using Cramers Rule to Solve a Linear System

(1/2) Use Cramers rule to solve

Solution:

832

30643

62

321

321

31

!

!

!

xxx

xxx

xx

-

!

-

!

-

!

-

!

821

3043

601

,

381

6303

261

328

6430

200

,

321

643

201

32

1

AA

AA

E l


112/112

Example

Using Cramers Rule to Solve a Linear System

(2/2) Therefore

11

38

44

152

)det(

)det(

1118

4472

)det()det(,

1110

4440

)det()det(

33

22

11

!!!

!!!

!

!!

A

Ax

AAx

AAx

linear algebra(ppt)updated

Documents