induction matrix

8/8/2019 Induction Matrix

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Algebra


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Topics:

Matrix

Equality of Matrices

Types of Matrix

Matrix Operations

Determinants of Matrix

Carmers rules (Determinant Method)

Applications


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Matrix

A !

a11 ,~ , a1n

a21 ,~ , a2 n

~ ~ ~ ~am1 ,~ , amn

-

! A ij_ a

A rectangular arrangement of mn numbersinto m horizontal rows and n vertical

columns enclosed by a pair of brackets [],

such as


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Equality of Matrices

Two M tr ces d B re s d to be equ lwr tte =B f they re of the s e order d

f ll correspo d g e tr es re equ l

-

!

432

015A

-

2232

0132

xB


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Row Matrix

[1 x n] matrx

? A _ ajn aaaaA ,,21 !~

A Matrx thathasexactlyonerow iscalled RowMatrix


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Column Matrix

_ ai

m

a

a

aa

A 2

1

!

-

~!

[m x 1] matrix

A Matrix consistingofa singlecolumn iscalledColumn Matrix


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Square Matrix

B ! 5 4 7

3 6 1

2 1 3

-

Anm

Xnmatrix

is

said

to

be

square

matrix

ofordernifm=n

Samenumberofrowsandcolumns


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Identity Matrix

I !

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

-

ASquare

matrix

is

said

to

be

identity

matrix

orunitmatrixifallitsmaindiagonal

elementsare 1sandallotherentriesare 0s.


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Triangular Matrix

A square matrix issaidtobe an upper(lower)triangularmatrix ifalltheelements

below(above)the main diagonal arezeros.

-

!

300

160

745

B

-

!

342061

005

C


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Diagonal Matrix

A square matrix issaidtobediagonal ifeachofitsentries notfallingon the main

diagonal iszero.

-

!

300

000005

B


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Scalar Matrix

A diagonal matrix whoes allthediagonalelements areequal iscalled a Scalarmatrix

-

!

500

050005

B


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Transpose Matrix

A' !

a11 a21 ,~, am1

a12 a22 ,~, am 2

~~~~~a1n a2n ,~, amn

-

Rows become columns and

columns become rows


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Matrix Operations Addition

Subtraction

Multiplication Inverse


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Addition

1

4

2

3

5

8

6

7+ =

A B+ =


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Addition ConformabilityTo add two matrices A and B:

# of rows in A = # of rows in B

# of columns in A = # of columns in

B

TwoMatricesofsameorder


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Subtraction

1

4

2

3

5

8

6

7 - =

B A- =


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Subtraction

1

4

2

3

5

8

6

7 - =

4

4

4

4

B A- = C


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Subtraction Conformability

To subtract two matrices A and B:

# of rows in A = # of rows in B

# of columns in A = # of columns in

B

TwoMatricesofsameorder


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Problem: 1

An automobile dealer sells two car models,

standard and deluxe. Each is available in one oftwo colors, white and red. His sales for the monthsof January & February are given by the matrices

Jan:

Feb:

Find his total sales for each model and color for bothmonths.

Standard Deluxe

White 2 1

Red 3 4

Standard Deluxe

White 3 1

Red 2 3


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Problem 2:

Three persons buy cold drinks of different

brands A, B & C. The first person buys 12bottles of A, 5 bottles of B & 3 bottles of C.The second person buys 4 bottles of A, 6bottles of B & 10 bottles of C. The third

person buys 6 bottles of A, 7 bottles of B &9 bottles of C. Represent these informationin the form of a matrix. If each bottle ofbrand A costs Rs.4, Each of B costs Rs. 5and each of C costs Rs. 6, then using matrix

operations. Find the total sum of moneyspent individually by the three persons forthe purchase of cold drinks.


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Multiplication Conformability Regular Multiplication

To multiply two matrices A and B:

# of columns in A = # of rows in B

Multiply: A (m x n) by B (n by p)


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Multiplication General Formula

Cij= 'A

ikx B

kjk=1

n


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MultiplicationI

1

4

2

3

5

8

6

7 x =

A Bx =


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MultiplicationII

1

4

2

3

5

8

6

7x =

A Bx = C

(5x1)

C11

='A11

xB11k=1

n


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MultiplicationIII

1

4

2

3

5

8

6

7x =

A Bx = C

(5x1)+(6x3)

C11

='A12

xB21k=2

n


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MultiplicationIV

1

4

2

3

5

8

6

7x =

A Bx = C

23 (5x2)+(6x4)

C12

='A1k

xBk2k=1

n


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MultiplicationV

1

4

2

3

5

8

6

7x =

A Bx = C

23

(7x1)+(8x3)

34

C21

='A2k

xBk1k=1

n


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MultiplicationVI

1

4

2

3

5

8

6

7x =

A Bx = C

23 34

(7x2)+(8x4)31

C22

='A2k

xBk2k=1

n


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MultiplicationVII

1

4

2

3

5

8

6

7x =

A Bx =

C

23 34

31 46

m x n n x p m x p


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DETERMINANTS OF A 3 X 3 MATRIX


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1 1 1

2 2 2

3 3 3

a b c

a b c

a b c

= +2 2

1

3 3

b ca

b c

1 1

2

3 3

b ca

b c

1 1

3

2 2

b ca

b c

= 1 2 3 3 2

a b c b c + 2 1 3 3 1

a b c b c 3 1 2 2 1

a b c b c

= 1 2 3 3 2a b c b c + + 2 1 3 3 1( 1)a b c b c 3 1 2 2 1a b c b c

=1 1

a A + +2 2

a A3 3

a A


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2 35

1 4

5 1 2

3 2 3

8 1 4

=


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2 35

1 4

5 1 2

3 2 3

8 1 4

=1 2

( 3)1 4


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2 35

1 4

5 1 2

3 2 3

8 1 4

=1 2

( 3)1 4

1 28

2 3

+


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2 35

1 4

5 1 2

3 2 3

8 1 4

=1 2

( 3)1 4

1 28

2 3

+

=

5 8 ( 3)

( 3) 4 2 +

8 3 ( 4)


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2 35

1 4

5 1 2

3 2 3

8 1 4

=1 2

( 3)1 4

1 28

2 3

+

=

5 8 ( 3)

( 3) 4 2 +

8 3 ( 4)

= 55 ( 6) + 56


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2 35

1 4

5 1 2

3 2 3

8 1 4

=1 2

( 3)1 4

1 28

2 3

+

=

5 8 ( 3)

( 3) 4 2 +

8 3 ( 4)

= 55 ( 6) + 56

= 117


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Problem 2:

The annual sale volumes of three products X, Y, Zwhoes sales prices per unit are Rs.3.50, Rs.2.75,Rs.1.50 respectively, in two differentmarkets I &II are shown below:

Product

Market X Y Z

I 6000 9000 13000

II 12000 6000 17000

Find the total revenue in each market with the helpof matrices.


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Unary Operations: Inverse

Inverse of (2 x 2) matrix

Find determinant

Swap a11 and a22

Change signs of a12 and a21 Divide each element by determinant

Check by pre- or post-multiplying by

inverse


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Inverse of 2 x 2 matrix

Find the determinant

= (a11 x a22) - (a21 x a12)

For

det(A) = (2x3) (1x5) = 1

2

3

5

1=A


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Swap elements a11 and a22Thus

becomes

2

3

5

1=A

3

2

5

1


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Change sign of a12 and a21Thus

becomes

3

2

5

1=A

3

2

-5

-1


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Divide every element by the determinant

Thus

becomes

(luckily the determinant was 1)

3

2

-5

-1=A

3

2

-5

-1


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Check results with A-1A = I

Thus

equals

3

2

-5

-1x

1

1

0

0

2

3

5

1


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CRAMERSRULE:

Its an application of determinants is to solve a

system of linear equations in which number ofvariables are equal to the number of equationsand the coefficient matrix of the system of

equations is non-singular


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Problem 3:

a) X - 3Y + 4Z = 3

2X 5Y + 7Z = 6

3X 8Y + 11Z = 11

induction matrix

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