DEFINITION OF MATRIX

A MATRIX is an ordered rectangular array of numbers or

functions.

The numbers or functions are called elements or the entries of

the matrix.

Example of a MATRIX

• A =

1 2 3

4 5 6

7 8 9

ORDER OF A MATRIX = ROW * COLUMN

= ( m * n )

For the above example order of A matrix = 3 * 3

Elements of A matrix = 1,2,3,4,5,6,7,8,9.

TYPES OF MATRICES

Rectangular matrix 1 2 3 4

5 6 7 8

Column matrix A = 1

2

Row matrix A = 1 2 3

2 * 4

A

=

2 * 1

1 * 3

Square matrix A = 0 4

6 3

Diagonal matrix A = 1 0 0

0 5 0

0 0 7

Scalar matrix A = 2 0 0

0 2 0

0 0 2

2 * 2

3 * 3

3 * 3

Identity matrix A = 1 0 0

0 1 0

0 0 1

Null matrix A = 0 0

0 0

Triangular matrix A = 1 2 3

0 4 5

0 0 6

3 * 3

2 * 2

3 * 3

(can be upper or lower triangular matrix.)

OPERATIONS ON MATRICES

Addition / Subtraction : When two matrices are added or subtracted then the order of matrix should be same.

for e.g.- A + B = C

1 2 0 5 1 7

3 4 7 8 10 12

Same is the procedure for Subtraction of matrices.

+ =

Multiplication :

1) with scalar- e.g. A = 1 4

7 4

2A = 2 1 7 = 2 14

7 4 14 8

2)Matrix multiplication

Necessary condition for matrix multiplication:

1.Column of first matrix should be equal to the row of second of matrix.

2 * 2

2 * 22 * 2

e.g.- 2 3 1 2

4 0 0 0

2*1+3*0 2*2+3*0

4*1+0*0 4*2+0*0

2 4

4 8

2 * 2

2 * 2

2 * 2

*

TRANSPOSE OF A MATRIX

• A matrix obtained by interchanging the rows and columns of the original matrix. It is denoted by A`.

E.g.- 1 2 ; A` = 1 3

3 0 2 0

A =

DETERMINANT OF A MATRIXTo every square matrix of associate a

number (real or complex) called determinant of the matrix.

REMARKS:

It is denoted by modulus sign i.e. A .

Only square matrices have determinants.

e.g.-for a matrix of order two

A = 1 2 : A = 1*1 – 2*2

2 1 = 1 - 4 => -3

• e.g.- for a matrix more than two

1 2 4

-1 3 0

4 1 0

A = 4 -1 3 - 0 1 2 +0 1 2

4 1 4 1 -1 3

= 4 (-1-12) – 0+0 = -52

A =

• Adjoint of a matrix : The adjoint of a square matrix A = [aij] n * n is defined as the transpose of the matrix [Aij] n * n where Aij is the cofactor of the element aij. It is denoted by adj A.

• Inverse of a matrix :

The inverse of an inverse matrix itself

The transpose of the inverse of a matrix is

equal to the inverse of the transpose.

The inverse of a matrix, if it exists, is unique.

A = AdjA

A

-1

INRODUCTION TO APPLICATION OF MATRICES

Matrices are one of the most powerful tools in Maths. The evolution of concept of matrices is the result of a n attempt to obtain compact and simple methods of solving system of linear equation. Matrix notation and operations are used in ELECTRONIC SPREADSHEET, programs for personal computer which in turn is used in different areas of business and science like BUDGETING, SALES PROJECTION, COST ESTIMATION etc. Also many physical operations such as magnifications, rotations and reflection through a plane can be represented mathematically by matrices. This mathematical tool is not only used in certain branches of sciences but also in genetics, economics, sociology, modern psychology and in industrial management.

APPLICATION TO MATRICES• Two-commodity market equilibrium x1p1+y1p2=z1 or x1 y1 p1 = z1

x2p1+y2p2=z2 x2 y2 p2 = z2

A= a1 b1 A=a1b2-a2b1

a2 b2 Also C= b2 - a2 A = 1 b2 - b1

-b1 a1 a1b2-a2b1 -a2 a1

Further, p1 = 1 b2 -b1 z1

p2 a2b2- a2b1 -a2 -a1 z2

Thus equilibrium prices are p1=z1b1-z2b2

a1b2-a2b2

And p2=z2a1-z1a2 a1b2-a2b1

Y=C+I

C=a+bY

1 -1 Y I

-b 1 C a

I -1 I+a C = 1 I a+bI

a 1 I-b -b a = 1-b

1 -1 1 – b

-b 1

NATIONAL INCOME METHOD

=

Y = =

e.g.Amount of 4,000 n three investments @ 7% , 8%, and 9% p.a. resp. The total

annual income is rs. 317.50 and the annual income from the first investment is rs.5 more than the income from the second. Find the amount of each investment.

Solution

1 1 1 a 4000

7 8 9 b = 31750

7 -8 0 c 500

A = 23; A = 34500; A = 28750 ; A = 28750;

1st investment = 34500 ; 2nd investment = 28750 ;

23 23

3rd investment = 28750

23

MARKOV BRAND-SWITCHING MODEL

• There are 2 brands A and B. Let the current market share of brand A be 60% and that of B be 40%.Since brand switching takes place so 70% of the consumers of brand A continue to use it while remaining 30% switch to brand B. Similarly, 80% of the consumers of brand B continue to use it while remaining 20% switch to brand A.

S = [0.6 0.4]

P = 0.7 0.3

0.2 0.8A

B

S(1) = [0.6 0.4] 0.7 0.3

0.2 0.8 = [0.5 0.5]

S(2) = S(1).P S[ I – P ] = 0

[SA SB] 1 – 0.7 -0.3 0

- 0.2 1 – 0.8 = 0

[SA SB] 0.3 -0.3 0 -0.2 0.2 = 0 SA + SB = 1 0.3SA – 0.2SB = 0We get SA = 0.4 or 40% and SB = 0.6 or 60%

• In many practical situations additional information about the matrices involved is known. An important case are sparse matrices, i.e. matrices most of whose entries are zero. There are specifically adapted algorithms for, say, solving linear systems Ax = b for sparse matrices A, such as the conjugate gradient method.[31]

• An algorithm is, roughly speaking, numerical stable, if little deviations (such as rounding errors) do not lead to big deviations in the result. For example, calculating the inverse of a matrix via Laplace's formula (Adj (A) denotes the adjugate matrix of A)

• A-1 = Adj(A) / det(A)

• Chemistry makes use of matrices in various ways, particularly since the use of quantum theory to discuss molecular bonding and spectroscopy. Examples are the overlap matrix and the Fock matrix using in solving the

Roothaan equations to the molecular orbitals of the Hartree–Fock method.

• Geometrical optics provides further matrix applications. In this approximative theory, the wave nature of light is neglected. The result is a model in which light rays are

indeed geometrical rays.

• If the deflection of light rays by optical elements is small, the action of a lens or reflective element on a given light ray can be expressed as multiplication of a two-component vector with a two-by-two matrix called ray transfer matrix: the vector's components are the light ray's slope and its distance from the optical axis, while the matrix encodes the properties of the optical element. The matrix characterizing an optical system consisting of a combination of lenses and/or reflective elements is simply the product of the components' matrices.

THERE ARE MANY MORE APPLICATIONS TO MATRICES IN DAY-TO-DAY LIFE ALSO.

LIMITATIONS OF MATRICES

Complicated calculations.

Difficulty in finding DETERMINANT of a 4 * 4 matrix and more.

Time consuming.

Inappropriate and doubtful results.

Lengthy procedure involved.

Tends to create confusion which increases the proportion of mistakes.

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• ARPITA LATTA• SAKSHI SRIVASTAVA• MAHIMA SHANKAR• ARUSHI SINGH• SAYANTI SANYAL• RAHUL ANAND• PALLAVI SHUKLA• SOUMYA R.• VAIBHAV TYAGI