1 matrices and determinants sjk

8/10/2019 1 Matrices and Determinants Sjk

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Lecture 1

Matrices and Determinants

MO 091204 -Mathematic Engineering

Ocean Engineering ITS


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1.1 atrices

1.2 Operations of matrices

. ypes o matr ces

1.4 Pro erties of matrices1.5 Determinants

1.6 Inverse of a 33 matrix


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1.1 Matrices

2 3 7A = 2 1 4B

=

4 7 6

.

rectangular array of numbers enclosed by a pair of bracket.

Why matrix?


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1.1 Matrices

7,+ =x y

It is easy to show thatx = 3 andy = 4.

3 5. =x y

2 7,+ = x y z

How about solving2 4 2,

5 4 10 1,

=

+ + =

x y z

x y z

3 6 5. = x y z


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11 12 1 K na a a

1.1 Matrices

21 22 2 =

M On

a a aA

1 2m m mn

numbers ai are called elements. First subscript indicates therow; second subscript indicates the column. The matrix

consists of mn elements

It is called the m n matrixA = [aij] or simply the matrixA if number of rows and columns are understood.


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S uare matrices

1.1 Matrices

When m = n, i.e.,11 12 1

K na a a

a a a

1 2

=

M O

n

n n nn

A

a a a

A is called a square matrix of order n or n-square

matrix

elements a11, a22, a33,, ann called diagonal elements.

....

11 221

+ + +

=

=

n

ii nni a a a a


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E ual matrices

1.1 Matrices

Two matricesA = [a ] andB = [b ] are said to be e ual(A =B) if each element ofA is equal to the corresponding

element ofB, i.e., aij = bij for 1 i m, 1 j n.

ifpronouns if and only if

, ij ij , ;if aij = bij for 1 i m, 1 j n, it impliesA =B.


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E ual matrices

1.1 Matrices

1 0 =

a bB

=

Given thatA =B, find a, b, c and d.

4 2 c d

ifA =B, then a = 1, b = 0, c = -4 and d= 2.


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Zero matrices

1.1 Matrices

Ever element of a matrix is zero, it is called a zero matrix,

i.e.,

0 0 0 K0 0 0 =

M OA


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Sums of matrices

. pera ons o ma r ces

IfA = [aij] andB = [bij] are m n matrices, thenA + Bs e ne as a ma r x = , w ere = ij , ij = ij

+ bij for 1 i m, 1 j n.

0 1 4=

A

1 2 5=

BExample: if and

EvaluateA + B andA B.

1 2 2 3 3 0 3 5 3

0 ( 1) 1 2 4 5 1 3 9

+ + + + = = + + +

A B

1 2 2 3 3 0 1 1 3

0 ( 1) 1 2 4 5 1 1 1

= = A B


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Sums of matrices


Two matrices of the same order are said to be

or a on or su rac on.

Two matrices of different orders cannot be added or

su tracte , e.g.,

2 3 7 1 3 1

1 1 5 4 7 6

are con orma e or a t on or su tract on.


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Scalar multi lication


Let be any scalar andA = [aij] is an m n matrix. Then= ij or , , .e., eac e emen n

A is multiplied by .

0 1 4=

AExample: . Evaluate 3A.

3 3 0 3 1 3 4 0 3 12= = A

, , . ., ij .

negative ofA. Note:

A = 0 is a zero matrix


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MatricesA,B and C are conformable,

A + B = B + A

A + B +C = A +B +C

(commutative law)

associative law

(A +B) = A + B, where is a scalar

an you prove t em


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=

Let C=A +B, so cij = aij + bij.

.

Consider cij = (aij + bij ) = aij + bij, we have, C=

+

Since C = (A +B), so (A +B) = A + B


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Matrix multi lication


IfA = [aij] is a m p matrix andB = [bij] is ap nma r x, en s e ne as a ma r x = ,

where C= [cij] with

p

1 1 2 21

...=

ij ik kj i j i j ip pj

k

1 2 3 =A

1 2

2 3

=BExam le: , and C = AB.

, .

5 0 Evaluate c21.

1 2

2 30 1 4

5 0

21 0 ( 1) 1 2 4 5 22= + + =c


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1 2 3 =

1 2 =

0 1 4 5 0

, , .

1 1 2 2 3 5 18c = + + =

12

21

1 21 2 2 3 3 0 81 2 3

2 30 ( 1) 1 2 4 5 220 1 4

c

c

= + + = = + + =

22 0 2 1 3 4 0 3c = + + =

1 2

2 30 1 4 22 3

5 0

C AB= = =


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In particular,A is a 1 m matrix andB is a mma r x, .e.,

[ ]11 12 1...= mA a a a

11

21

= M

b

B

1 mb

then C = AB is a scalar.

1 1 11 11 12 21 1 1...= = + + +m

k k m mC a b a b a b a b


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BUTBA is a m m matrix!

11 11 11 11 12 11 1

21 21 11 21 12 21 1...

= =

K m

m

b b a b a b a

b b a b a b aBA a a a

1 1 11 1 12 1 1

m m m m mb b a b a b a

SoAB BA in general !


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Pro erties:


MatricesA,B and C are conformable,

A(B + C) = AB + AC

A + B C = AC+BC

A(BC) = (AB) C

AB BA in general

AB = 0 NOT necessaril im l A = 0 orB = 0

AB = ACNOT necessarily implyB = C


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. ypes o matr ces

Identity matrix

The inverse of a matrix

The transpose of a matrix

ymmetr c matr x

Orthogonal matrix


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1.3 Types of matrices

= >

Identity matrix

,

called upper triangular, i.e., 11 12 1

22 20

K n

n

a a a

a a

0 0

M O

nna

A square matrix whose elements aij = 0, for i


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=

Identity matrix

, . ., , , . .,

11 0 0

0 0

Ka

a

0 0

=

M O

nn

D

a

11 22diag[ , ,..., ]= nnD a a a

is called a diagonal matrix, simply


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Identity matrix

In particular,a

11= a

22= = a

nn= 1

, the matrix iscalled identity matrix.

Properties:AI = IA = A

1 0 0 Examp es o i entity matrices: an

0 1 0 1 0

0 0 1


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Special square matrix

. ,

andB such thatAB = BA, thenA andB are said to becommute.

Can you suggest two matrices that must commute with asquare ma r xAns:Aitself,theidentitymatrix,..

IfA andB such thatAB = -BA, thenA andB are said to

e ant -commute.


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= =

e nverse o a matr x

,

called the inverse ofA (symbol:A-1); andA is called theinverse ofB s mbol:B-1 .

6 2 3

1 1 0B

=

1 2 3

1 3 3A

=Exam le:1 0 1

ShowB is the the inverse of matrixA.

1 2 4

1 0 0

0 1 0AB BA

= =Ans: Note that0 0 1 Can you show the

details?


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e transpose o a matr x

columns of a matrixA is called the transpose ofA (writeAT).

Example:1 2 3

4 5 6

=

A

1 4 The transpose ofA is 2 5

3 6

=

TA

For a matrixA = [aij], its transposeAT= [bij], where bij= aji.


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ymmetr c matr x

T= = , . .,

aij for all i andj.

mus e symme r c. y

1 2 3

2 4 5

= .3 5 6

T

-

- ,i.e., aji = -aij for all i andj.

- must e s ew-symmetric. W y?


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Orthogonal matrix

T= T = T = , . .,

A-11/ 3 1/ 6 1/ 2

Example: prove that is orthogonal.1/ 3 2 / 6 0

1/ 3 1/ 6 1/ 2

=

A

Since, . Hence,AAT= ATA = I.1/ 3 1/ 3 1/ 3

1/ 6 2 / 6 1/ 6TA

=

1/ 2 0 1/ 2 Can you show thedetails?

Well see that orthogonal matrix represents a

rotation in fact!


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1.4 Properties o matrix

(AB

)

-1 = B-1A-1

AT T= A and A T= AT

(A + B)T= AT + BT

(AB)T= BTAT


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1.4 Properties o matrix

-1 = -1 -1 .

Since (AB) (B-1A-1) = A(B B-1)A-1 = Iand

(B-1A-1) (AB) = B-1(A-1A)B = I.

-1 -1 , .


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1.5 Determinants

Determinant of order 2

Consider a 2 2 matrix: 11 12a a

A =

21 22

, ,

evaluated by

11 1211 22 12 21

21 22

| | a a a a a aa a= =


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Determinant of order 2

1.5 Determinants

easy to remember (for order 2 only)..

11 12

11 22 12 21

21 22

| |a a

A a a a a

a a

= =+

1 2

-

3 4

1 21 4 2 3 2= =


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1.5 Determinants

order.

. every e ement o a row co umn s zero,

e.g., , then |A| = 0.

1 2

1 0 2 0 0= =

T

determinant of a matrix = thatof its trans ose.

3. |AB| = |A||B|


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1.5 Determinants

matrix is either +1 or1. T , .

Since|AAT| = |A||AT | = 1

and|AT| = |A|

, so|A|2 = 1

or|A| =

.


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1.5 Determinants

11 12a a

21 22a a

Its inverse can be written as 22 121 1a a

A

=

21 11a a

1 0 =1 2

The determinant of A is -2

Hence, the inverse of A is 1 1 01/ 2 1/ 2A =

How to in an inverse or a 3x3 matrix?


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1.5 Determinants of order 31 2 3

Consider an example: 4 5 6

7 8 9

A=

Its determinant can be obtained by:

4 5 1 2 1 24 5 6 3 6 9

7 8 7 8 4 57 8 9

A= = +

( ) ( ) ( )3 3 6 6 9 3 0= + =

ou are encourage o n e e erm nan y us ng o er

rows or columns


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1.6 Inverse of a 33 matrix1 2 3

Cofactor matrix of 0 4 5

1 0 6

A=

The cofactor for each element of matrix A:

4 5 0 5 0 4 110 6

121 6

131 0

2 3 1 3 1 221

0 6 22

1 623

1 0

2 3 1 3 1 231 4 5

= = 320 5

= = 33 0 4= =


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1.6 Inverse of a 33 matrix

Cofactor matrix of is then given by:0 4 51 0 6

A

=

12 3 2

2 5 4


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1.6 Inverse of a 33 matrix1 2 3

Inverse matrix of is given by:0 4 5

1 0 6

A=

1

24 5 4 24 12 21 1

12 3 2 5 3 5

T

A

= = 2 5 4 4 2 4

5 22 3 22 5 22

=

1 matrices and determinants sjk

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