Presentation on matrix

Presented by
Masuda Mahbub
Nahin Mahfuz seam
Saddique Muhammad Takbir Dakhin
Dhaka school of Economics
University of Dhaka
Presentation on
Matrix and it`s aplication

 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
aa
aa
,,
,,
,,
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 


















0
AallforThen,zero.allareentrieswhose0matrixThe
11212
842
156
701
076
143
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)












01412
286
076
143
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.IBB,matrixmnany
forandAAIA,matrixnmanyfor
100
010
001
ImatrixIdentity
.
Example
....)...( 2211
2
1
21





























































nn
mjimjiji
mj
j
j
imii
fDeBfCeA
dDcBdCcA
bDaBbCaA
DC
BA
fe
dc
ba
bababa
b
b
b
aaa





If A has dimensions k × m and B has dimensions m × n, then the product
AB is defined, and has dimensions k × n. The entry (AB)ij is obtained
by multiplying row i of A by column j of B, which is done by multiplying
corresponding entries together and then adding the results i.e.,
5

BC.ACB)CAC, (AABC)A(B
ABBA
A(BC).C, (AB)CB)(AC)(BA



:LawsveDistributi
:AdditionforLaweCommutativ
:LawseAssociativ
The matrix addition, subtraction, scalar multiplication and matrix
multiplication, have the following properties.
6

















2313
2212
2111
232221
131211
aa
aa
aa
aaa
aaa
T
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

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
bcad
dc
ba






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-1A
The formula for deriving the inverse is
10

Calculation of Inversion using
Determinants
2 4 5
0 3 0
1 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 3
3, 0, 3,
0 1 1 1 1 0
4 5 2 5 2 4
4, 3, 4,
0 1 1 1 1 0
4 5 2 5 2 4
15, 0, 6,
3 0 0 0 0 3
det 9,
3 4 15
0 3 0 .
3 4 6
3
1
,
9
C C C
C C C
C C C
A
C C C
adjA C C C
C 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 b
a x a x a x a x b
a x a x a x a x b
    
    
    
K
K
K K K K K K
K
The system of linear equations:
can be rewritten as the matrix equation Ax=b, where
1 1
11 1
2 2
1
, , .
n
k kn
n k
x b
a a
x b
A x b
a a
x b
   
     
            
          
   
K
M O M
M M
L
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

1
-1
Matrix Inversion
4 1 2 x 4
5 2 1 ; X y ; b 4
1 0 3 z 3
6 -3 -3
1
A -14 10 6
6
-2 1 3
x 6 -3 -3 4
1
y -14 10 6 4
6
z -2 1 3 3
1 2; y 1 3; z 5 6
AX d
A
X A b
x


     
            
          

 
   
  
     
          
          
  
4 2 4
5 2 4
3 3
x y z
x y z
x z
  

  
  
13

 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

 Field of Geology
● Taking seismic surveys
● Plotting graphs & statistics
● Scientific analysis
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

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