Matrix & Determination ppt

1. 1
Matrices and Determinants
By
Nutan Prakash
Asstt. Professor
Department of Biotechnology
Shree M. & N. Virani Science College

2. Contents
• Matrices
• Order of Matrix
• Operations of matrices
• Types of matrices
• Properties of matrices
• Determinants
• Inverse of a 3×3 matrix
2

3. 3
2 3 7
1 1 5
A
 
=  − 
Matrices
1 3 1
2 1 4
4 7 6
B
 
 =  
  
A matrix is a rectangular array of numbers enclosed
by a pair of bracket.
11 12 1
21 22 2
1 2
 
 
 =
 
 
 
K
M O
n
n
m m mn
a a a
a a a
A
a a a
In the matrix
numbers aij are called elements. First subscript indicates the row;
second subscript indicates the column.
The matrix consists of mn elements. It is called “the m × n matrix

4. Order of Matrix
A matrix having m rows and n columns is called
a matrix of order m × n or simply m × n matrix
(read as an m by n matrix).
4
Determine the size of the matrix shown below?
The matrix has 3 rows and 2 columns. Its size is 3 x 2.
2 1
7 -4
3 1
 
 =  
  
B

5. 5
Determine the size of the matrix
shown below?
The matrix has 2 rows and 3 columns. Its
size is 2 x 3.
2 -3 5
-1 4 6
 
=  
 
A

6. 6
Determine the size of the matrix
shown below.
The matrix has 3 rows and 3 columns. Its
size is 3 x 3. It is a square matrix.
2 -1 3
4 6 1
-5 2 1
 
 =  
  
C

7. If a matrix has 8 elements, what are the possible orders it
can have?
Soln.: We know that if a matrix is of order m × n, it has mn
elements. Thus, to find all possible orders of a matrix with 8
elements, we will find all ordered pairs of natural numbers,
whose product is 8.
 Thus, all possible ordered pairs are (1, 8), (8, 1), (4, 2),
(2, 4)
 Hence, possible orders are 1 × 8, 8 ×1, 4 × 2, 2 × 4
7

8. 8
Transpose
 The transpose of a matrix A is denoted as A’ and is obtained by
interchanging the rows and columns.
 Thus, if A has a size of m x n, A’ will have a size of n x m.
 If the transpose operation is applied twice, the original matrix is
restored.
Determine the transpose of the matrix A below.
2 -3 5
-1 4 6
 
=  
 
A
2 -1
-3 4
5 6
 
 =  
  
A'

9. Types of Matrices
A matrix having only one row is called a row matrix.
A matrix having only one column is called a column
matrix.
A matrix of either form is called a vector.
The row matrix is called a row vector and the column
matrix is called a column vector.
9

10. 10
Express the integer values of time from 0
to 5 s as a row vector.
The size of the vector is 1 x 6.
[0 1 2 3 4 5]=t

11. 11
Express the variables x1, x2, and x3 as
a column vector.
1
2
3
x
x
x
 
 =  
  
x

12. Square matrix
• A matrix in which the number of rows are equal to the number of
columns, is said to be a square matrix.
• Thus an m × n matrix is said to be a square matrix if m = n and
is known as a square matrix of order ‘n’.
12
2 -1 3
For Example 4 6 1 is a square matrix of order 3.
-5 2 1
 
 =  
  
C

13. Diagonal matrix
A square matrix is said to be diagonal matrix of all
non diagonal elements are zero.
13
-1 0 0
For Example A 0 2 0 is a diagonal matrix of order 3.
0 0 3
 
 =  
  

14. Scalar matrix
A diagonal matrix is said to be a scalar matrix if its
diagonal elements are equal, that square matrix is
said to be a scalar matrix.
14
-1 0 0
1 0
For Ex: A=[3], B= , C 0 -1 0 are
0 1
0 0 -1
scaler matrices of order 1,2 and 3 respectively.
 
−   = ÷  −    

15. Identity matrix
A square matrix in which elements in the diagonal are
all 1 and rest are all zero is called an identity matrix.
15
1 0 0
1 0
For Ex: A=[1], B= , C 0 1 0 are
0 1
0 0 1
identity matrices of order 1, 2 and 3 respectively.
 
   = ÷  
    

16. Zero matrix
A matrix is said to be zero matrix or null matrix if all its
elements are zero.
16
0 0 0
0 0
For Ex: A=[0], B= , C 0 0 0 are
0 0
0 0 0
all zero matrix, it is denoted by 0.
 
   = ÷  
    

17. 17
Symmetric matrix
A matrix A such that AT
= A is called
symmetric, i.e., aji = aij for all i and j.
A + AT
must be symmetric. Why?
Example: is symmetric.
1 2 3
2 4 5
3 5 6
 
 = − 
 − 
A
A matrix A such that AT
= -A is called skew-
symmetric, i.e., aji = -aij for all i and j.
A - AT
must be skew-symmetric. Why?

18. 18
Two matrices A= [aij] and B = [bij] are said to be equal
if
(i)they are of the same order
(ii)each element of A is equal to the corresponding
element of B, that is aij = [bij] for all i and j.
EQUALITY OF MATRICES
2 3 2 3
For Ex: and are equal matrices but
0 1 0 1
3 2 2 3
and are not equal matrix.
0 1 0 1
Symbolically if two matrices are equal we write A=B.
   
 ÷  ÷
   
   
 ÷  ÷
   

19. 19
Given that A = B, find a, b, cand d.
1 0
4 2
A
 
=  − 
a b
B
c d
 
=  
 
Example: and
if A = B, then a= 1, b= 0, c= -4 and d= 2.

20. 20

21. 21
Sums of matrices
1.2 Operations of matrices
If A = [aij
] and B = [bij
] are m × n matrices,
then A + B is defined as a matrix C = A + B,
where C= [cij
], cij
= aij
+ bij
.
1 2 3
0 1 4
 
=  
 
A
2 3 0
1 2 5
 
=  − 
BExample: if and
Evaluate A + B and A – B.
1 2 2 3 3 0 3 5 3
0 ( 1) 1 2 4 5 1 3 9
+ + +   
+ = =   + − + + −   
A B
1 2 2 3 3 0 1 1 3
0 ( 1) 1 2 4 5 1 1 1
− − − − −   
− = =   − − − − − −   
A B

22. 22
 Two matrices of the same order are said to be
co nfo rm able for addition or subtraction.
 Two matrices of different orders cannot be
added or subtracted, e.g.,
are NOT conformable for addition or subtraction.
2 3 7
1 1 5
 
 − 
1 3 1
2 1 4
4 7 6
 
 
 
  

23. 23
Multiplication of a matrix by a scalar
Let λ be any scalar and A = [aij
] is an m × n
matrix. Then λA = [λaij
] for 1 ≤ i ≤ m, 1 ≤ j ≤ n,
i.e., each element in A is multiplied by λ.
1 2 3
0 1 4
 
=  
 
AExample: . Evaluate 3A.
3 1 3 2 3 3 3 6 9
3
3 0 3 1 3 4 0 3 12
× × ×   
= =   × × ×   
A
In particular, λ = −1, i.e., −A = [−aij
]. It’s called
the negative of A. Note: Α − A = 0 is a zero matrix

24. 24
Properties of matrices
Matrices A, B and C are conformable,
A + B = B + A
A + (B +C) = (A + B) +C
λ(A + B) = λA + λB,
where λ is a scalar
(commutative law)
(associative law)
(distributive law)

25. 25
Matrix multiplication
1 2 3
0 1 4
 
=  
 
A
1 2
2 3
5 0
− 
 =  
  
BExample: , and C = AB.
Evaluate c21
.
1 2
1 2 3
2 3
0 1 4
5 0
− 
   
   
    
21 0 ( 1) 1 2 4 5 22= × − + × + × =c
for multiplication of two matrices A and B, the number of
columns in A should be equal to the number of rows in B.
Furthermore for getting the elements of the product matrix, we
take rows of A and columns of B, multiply them element-wise
and take the sum.

26. 26
1.2 Operations of matrices
1 2 3
0 1 4
 
=  
 
A
1 2
2 3
5 0
− 
 =  
  
BExample: , , Evaluate C =
AB.
11
12
21
22
1 ( 1) 2 2 3 5 18
1 2
1 2 2 3 3 0 81 2 3
2 3
0 ( 1) 1 2 4 5 220 1 4
5 0
0 2 1 3 4 0 3
c
c
c
c
= × − + × + × =
−   = × + × + × =    ⇒     = × − + × + × =      = × + × + × =
1 2
1 2 3 18 8
2 3
0 1 4 22 3
5 0
C AB
− 
    = = =    
     

27. Properties of multiplication of matrices
1. The associative law: For any three matrices A, B and C.
We have (AB) C = A (BC), whenever both sides of the
equality are defined.
2. The distributive law For three matrices A, B and C.
(i)A (B+C) = AB + AC
(ii)(A+B) C = AC + BC, whenever both sides of equality
are defined.

28. 3. The existence of multiplicative identity For every
square matrix A, there exist an identity matrix of same
order such that IA = AI = A.
28

29. 29

30. Properties of Matrix
If matrices A, Band C are conformable
 A(B+ C) = AB+ AC
 (A+ B)C = AC + BC
 A(BC) = (AB) C
 AB≠ BAin general
 (AB)-1
= B-1
A-1
 (AT
)T
= A and (λA)T
= λ AT
 (A + B)T
= AT
+ BT
 (AB)T
= BT
AT
30

31. 31

32. 32

33. 33

34. Applications of Matrix Multiplication in Biology
Due to recent progress of DNA microarray technology, a
large number of gene expression profile data are being
produced. Matrix multiplication is used to analyze gene
expression in computational molecular biology.
Matrix multiplication is used in this technology to create
simple algorithms.
In the circulatory system, the red blood cells are constantly
being destroyed and replaced. Since they carry oxygen
throughout the body, their numbers are fixed. We can use
matrix multiplication to find out the level of red blood34

35. Matrix multiplication is used to find the frequency of sickle
cell allele of the gene for hemoglobin causes red blood
cells to collapse. Individuals with heterozygous genes for
sickle cell develop immunity against malaria, but
individuals homozygous for this gene tend to die at an
early age.
Matrix multiplication can be used to find out the frequency
of occurrence of this gene in individuals living in a certain
area and to calculate the possibility of this disease
happening in a progeny of a family. 35

36. Human populations have been increasing at a nearly
exponential rate over the last couple of thousand years.
Matrix multiplication is used for calculating population
expansion of a species, not just human beings, over a
period of time, provided it grows at a constant rate. This
can help monitor the population of an endangered or over-
populated species
36