matrix algebra : mathematics for business

LET’S START

WHAT IS MATRIX?

• > Rectangular presentation of symbols or numerical elements

• > Arranged systematically in rows and columns

• >Describes various aspects of a phenomenon which are inter-related in some manner.

These are some numbers.

1

4

2

3

Numbers can be organized in boxes, e.g.

1

4

2

3

Let’s take an example

• We may consider two linier equations,

2x - 3y + z = 7……. (1)4x + 5y - 3z = 5…… (2)

• Co-efficient of x, y, z in the equations-

In case of (1) are, 2, -3, 1

In case of (2) are, 4, 5, -3

So, (1)and (2) form this matrix Co-efficient matrix ,

2 x 3 matrix. (2 rows and 3 columns Matrix)

354

132

• After enclosing by two brackets, the rectangular array as well as the Matrix is treated as a single entry.

[ ]

Many Unorganized Numbers

28 39 57 17 38 18 38 65 10 73 16 73 77 63 18 56 18 74 82 20 10 75 84 19 47 14 11 84 08 47 57 58 49 48 88 84 47 48 43 05 61 75 98 47 32 98 15 49 01 38 65 43 17 65 21 79 43 17 59 41 37 59 43 17 97 65 41 35 75 49 03 86 93 41 76 73 19 57 75 49 27 59 34 27 59 43 19 74 32 17 43 92 65 94 13 75 93 41 65 99 13 47 75 83 47 48 73 98 47 39 28 17 49 03 63 91 40 35 42 31 87 49 75 48 91 37 59 13 48 75 94 13 75 45 43 54 75 48 90 37 59 37 59 43 75 90 33 57 75 89 43 67 74 34 92 76 90 34 17 34 82 75 98 34 27 69 31 75 93 45 13 59 84 76 59 13 47 69 43 17 91 34 75 93 41 75 90 34 15 74 91 35 79 57 42 39 57 49 02 35 74 23 57 75

Now it is a Matrix.

MATRIX DEFINED• A matrix is a rectangular array of numbers

arranged in rows and columns enclosed by a pair of brackets and subject to certain rules of presentation.

• A matrix can be enclosed by [ ] or || ||

Like,

Or,

354

132354

132

EXPRESSING MATRIX

• Matrix is denoted by a capital letter

• Elements are corresponded by small letters

• Small letters are followed by two suffixes m& n

• m represents the order of the Row

• n represents the order of the Column

Expressing Matrix (Cont.) Example,• A matrix having m rows and n columns

can be written as

nmmnmjmm

inijii

nj

nj

aaaaaaaaaaaaaaaa

A

21

21

222221

111211

TYPES OF MATRIX

Square Matrix

• A matrix, in which the number of rows is equal to the number of columns, is called a square matrix.

Square Matrix(cont.)

Example:

m x n matrix where m=n ; (m =row, n = column)

.

21

21

222221

111211

nmmnmjmm

inijii

nj

nj

aaaaaaaaaaaaaaaa

Row Matrix• A matrix having a single row is

called a row matrix.

Example:

[a…b…c…d]

Column Matrix• A matrix having a single column

is called a column matrix.

Example,

31

21

11

aaa

Diagonal Matrix• A square matrix all of whose

elements, except those in the leading diagonal, are zero is called a diagonal matrix.

Example:

nnaa

aA

000000

22

11

Unit Matrix• A scalar matrix of whose, diagonal

element is unity (or one) is called a Unit Matrix or an Identity Matrix. A Unit Matrix of order n is written as .

Example:

mI

100010001

4I

Zero Matrix or Null Matrix• A matrix, rectangular or square, each of

whose elements are zero is called a Zero Matrix or Null Matrix.

Denoted by 0.Example:

0000000000000000

0

Triangular Matrices

• A matrix where the elements are zero according to the superiority of the order of their rows (m) and columns (n) is called a Triangular Matrix.

Upper Triangular Matrix• If the order of row is greater than

the order of column of an element 0, then it is called as Upper Rectangular Matrix.

Example:

nmmn

inij

nj

nj

aaaaaaaaaa

A

00000

0 2222

111211

Lower Rectangular Matrix• If the order of row is smaller than

the order of column of an element 0, then it is called as Lower Rectangular Matrix.

Example:

nmaaaa

aaaaa

a

A

44434241

333231

2221

11

000000

Sub Matrix• A matrix obtained by deleting

some rows or columns or both of a given matrix is called a sub matrix of the given matrix.

Example:

From A= 2x2

1213

,123132213

33

Ato

Scalar Matrix• A square matrix when given in the form

of a scalar multiplication to an identity or unit matrix is called a Scalar Matrix.

• We can also say, a diagonal matrix whose all diagonal elements are equal is called a scalar matrix.

Example:

1001

00

aa

aaI

Symmetric Matrices• A symmetric matrix is a

special kind of square matrix A=[aij] for which,

jandiallforaa jiij

Symmetric Matrices (cont.)

• Example:

511162

125

Complex Conjugate of a Matrix

• It is a Matrix obtained by replacing all its elements by there respective complex conjugates.

Example:

735432

,735432

.ii

Athenii

Aif

Skew-symmetric Matrix• It is square matrix A if,

• The transpose of a square matrix is equal to negative of that matrix

• Or, we can say a square matrix A is called a Skew-symmetric Matrix if aij = - aij for all i and j. In a skew-symmetric matrix all the diagonal elements are zeroes.

AAt

Skew-symmetric Matrix(cont.)

• Example:

0660

A

Matrix Operatio

ns

Addition• Matrices can be added if only they

are of the same order.• The sum of two (mxn) matrices is

another matrix (mxn) whose elements are the sum of the corresponding elements in the component matrices.

Addition (cont.)• Example:

1

4

2

3

5

8

6

7+ =

A B+ =

1

4

2

3

5

8

6

7+ =

6

12

8

10

A B+ = C

Addition (cont.)• Example:

Subtraction• Matrices can be subtracted if only

they are of the same order. • The difference of two (mxn)

matrices is another matrix (mxn) whose elements are the difference of the corresponding elements in the component matrices.

Subtraction(cont.)• Example:

1

4

2

3

5

8

6

7- =

B A- =

1

4

2

3

5

8

6

7- =

4

4

4

4

B A- = C

Subtraction(cont.)• Example:

Multiplication• For matrix multiplication,

>the number of columns in the first matrix or vector must be equal to the number of rows in the second matrix or the vector.

Multiplication (cont.)• Example:

1

4

2

3

5

8

6

7x =

A Bx =

Multiplication (cont.)• Example:

1

4

2

3

5

8

6

7x =

A Bx = C

(5x1)+ (6x3)

C 11 = A 12 x B 21k=2

n

1

4

2

3

5

8

6

7x =

A Bx = C

23 (5x2)+ (6x4)

C 12 = A 1k x B k2k=1

n

Multiplication (cont.)• Example:

1

4

2

3

5

8

6

7x =

A Bx = C

23

(7x1)+ (8x3)

34

C 21 = A 2k x B k1k=1

n

Multiplication (cont.)• Example:

1

4

2

3

5

8

6

7x =

A Bx = C

23 34

(7x2)+ (8x4)31

C 22 = A 2k x B k2k=1

n

Multiplication (cont.)• Example:

1

4

2

3

5

8

6

7x =

A Bx = C

23 34

31 46

m x n n x p m x p

Multiplication (cont.)• Example:

Inverse of Matrix

• The operation of dividing one matrix directly by another doesn’t exist in matrix theory but equivalent of division of unit matrix by any square matrix can be accomplished (in most cases) by a process known as ‘Inversion of Matrix’.

• The concept of inverse matrix is useful in solving simultaneous equations, input output analysis and regression analysis.

Inverse of Matrix (cont.)

Solutions of simultaneous equations of matrix no.1

• Using matrices, calculate the values of x and y for the following simultaneous equations:

2x – 2y – 3 = 0 8 y = 7x + 2

Step 1:• Write the equations in the form ax + by = c

2x – 2y – 3 = 0 ⇒ 2x – 2y = 3 8y = 7x + 2 ⇒ 7x – 8y = –2

Step 2:• Write the equations in matrix form.

Step 3:• Find the inverse of the 2 × 2 matrix.

Determinant = (2 × –8) – (–2 × 7) = – 2

Step 4: Multiply both sides of the matrix

equations with the inverse.

So, x = 14 and y = 12.5

Solutions ofsimultaneous equations of matrixno 2:

• Now the aim is to combine the equations in such a way that we eliminate one of the variables.

Step 1:• If we multiply the first equation by

some number• then we can get the coefficient of

the "x" to be the same in both equations.

• Then we can subtract one equation from the other and the x-terms will cancel.

Step 2:• So let's multiply the first equation by 3,

so we have "6x" in both equations:

Step 4:• Thus we are left with an equation that

involves only "y", which we can therefore rearrange to see what the value of y is:

Step 3:• Now, as planned, we can subtract that

new equation from the second equation and the x-terms cancel

Step 5:• Having found y, we can substitute it into

either of the two original equations to find x. Let's substitute it into the first:

Step 6:• Rearranging that equation for

x we can see the value of x:

So our result is that

•X = 11 / 7•Y = 2 / 7.

APPLICATION OF MATRIX IN BUSINESS

• Let’s suppose that, we are working for the website travelocity.com to help people plan trips between various cities.

• Often the customers are business travelers so that they want to travel between cities in the morning to conduct a day’s business.

• Large cities often provide flights to many cities, but small cities often are quite limited in the number of cities that they service.

• The customers are particularly interested in travel between the following cities.

Adjacency matrices and Airlines

• The cities are Albany, Boston, New York, Philly, Wash, Richmond, Detroit, and Las Vegas.

• For simplicity, we will only use the first letter to refer to the city. Here is the flight information that we are given.

Adjacency matrices and Airlines

• From Boston there are flights to N, P, W, D• From Albany there are flights to N, W• From New York there are flights to B, P, W, R, D,

L• From Philly there are flights to N, B, W, R• From Wash there are flights to B, A, N, R, P, L• From Richmond there are flights to N, P, W• From Detroit there are flights to B, N• From Las Vegas there are flights to N, W

Adjacency matrices and Airlines

Now we can represent the flights and the routs in a very organized way using matrix.

Adjacency matrices and Airlines

Adjacency matrices and Airlines

From this chart, the passengers can easily find out their flights from one city to another.

Conclusion After the discussion given

above, we can conclude that,

• Matrix Algebra is a smart option to complete different business operations.

• This method can reduce our effort for preparing different calculations as well business predictions.