Presented byMasuda Mahbub

Nahin Mahfuz seamSaddique Muhammad Takbir Dakhin

Dhaka school of EconomicsUniversity of Dhaka

Presentation on Matrix and it`s aplication

Outline Definition of a Matrix Operations of Matrices Determinants Inverse of a Matrix Linear System Unique properties of matrix Uses of matrices

1

Matrix (Basic Definitions)

ij

knk

n

n

A

aa

aaaa

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1

221

111

A

Matrices are the rectangular agreement of numbers, expressions, symbols which are arranged in columns and rows.

2

Operations with Matrices (Sum,Difference)

AA

0A allfor Then, zero. all are entries whose0matrix The

1 12 1284 2

1 5 670 1

0 7 614 3

If A and B have the same dimensions, then their sum, A + B, is obtained by adding corresponding entries. In symbols, (A + B)ij = aij + bij . If A and B have the same dimensions, then their difference, A − B, is obtained by subtracting corresponding entries. In symbols, (A - B)ij = aij - bij .

3

Operations with Matrices (Scalar Multiple)

0 14 1228 6

0 7 614 3

2Example:

If A is a matrix and r is a number (sometimes called a scalar in this context), then the scalar multiple, rA, is obtained by multiplying every entry in A by r. In symbols, (rA)ij = raij .

4

Operations with Matrices (Product)

B.IB B,matrix mnany

for andA AI A,matrix n many for

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01 000 1

Imatrix Identity

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Example

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1

21

nn

mjimjiji

mj

j

j

imii

fDeBfCeAdDcBdCcAbDaBbCaA

DCBA

fedcba

bababa

b

b

b

aaa

If A has dimensions k × m and B has dimensions m × n, then the productAB is defined, and has dimensions k × n. The entry (AB)ij is obtainedby multiplying row i of A by column j of B, which is done by multiplyingcorresponding entries together and then adding the results i.e.,

5

BC. AC B)C AC, (A AB C) A(B

A B B A

A(BC). C, (AB)C B) (A C) (B A

:Laws veDistributi

:Additionfor Law eCommutativ

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Laws of Matrix Algebra

The matrix addition, subtraction, scalar multiplication and matrix multiplication, have the following properties.

6

OPERATIONS WITH MATRICES (TRANSPOSE)

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The transpose, AT , of a matrix A is the matrix obtained from A by writing its rows as columns. If A is an k×n matrix and B = AT then B is the n×k matrix with bij = aji. If AT=A, then A is symmetric

7

Example:

8

DETERMINANT OF MATRIX Determinant is a scalar

• Defined for a square matrix

• Is the sum of selected products of the elements of the matrix, each product being multiplied by +1 or -1

bcaddcba

det

9

INVERSE OF A MATRIX

Definition: An inverse matrix A-1 which can be found only for a square

and a non-singular matrix A ,is a unique matrix satisfying the relationship AA-1= I =A-1AThe formula for deriving the inverse is

10

Calculation of Inversion using Determinants

2 4 50 3 01 0 1

A

11 12 13

21 22 23

31 32 33

11 21 31

12 22 32

13 23 33

1

3 0 0 0 0 33, 0, 3,

0 1 1 1 1 0

4 5 2 5 2 44, 3, 4,

0 1 1 1 1 0

4 5 2 5 2 415, 0, 6,

3 0 0 0 0 3det 9,

3 4 150 3 0 .3 4 6

31,9

C C C

C C C

C C C

AC C C

adjA C C CC C C

So A

4 15

0 3 0 .3 4 6

Example: find the inverse of the matrix

Solve:

11

Systems of Equations in Matrix Form

11 1 12 2 13 3 1 1

21 1 22 2 23 3 2 2

1 1 2 2 3 3

n n

n n

k k k kn n k

a x a x a x a x ba x a x a x a x b

a x a x a x a x b

The system of linear equations:

can be rewritten as the matrix equation Ax=b, where

1 111 1

2 2

1

, , .n

k knn k

x ba a

x bA x b

a ax b

If an n×n matrix A is invertible, then it is nonsingular, and the unique solution to the system of linear equations Ax=b is x=A-

1b.12

EXAMPLE: SOLVE THE LINEAR SYSTEM

1

-1

Matrix Inversion

4 1 2 x 45 2 1 ; X y ; b 41 0 3 z 3

6 -3 -31A -14 10 66

-2 1 3

x 6 -3 -3 41y -14 10 6 46

z -2 1 3 31 2; y 1 3; z 5 6

AX d

A

X A b

x

4 2 45 2 4

3 3

x y zx y z

x z

13

Unique properties of matrices

In normal algebra , if we multiply two non-zero values, then the outcome will never be a zero . But if we multiply two non-zero values in matrix , then the outcome can be zero.

14

15

Field of Geology ● Taking seismic surveys

● Plotting graphs & statistics

● Scientific analysis

Application of MatrixIn our everyday life

16

Field of Statistics & Economics

● Presenting real world data such as People's habit, traits & survey data

● Calculating GDP

Field of Animation

● Operating 3D software & Tools

● Performing 3D scaling/Transforming

● Giving reflection, rotation

17

ANY QUESTIONS?