HOW TO UNDERSTAND RELATIONS AND FUNCTIONS ,INVERSE OF A FUNCTION

we are going to  discuss Relations and Functions , "How to understand  Relations and  Functions, Inverse of a Function" under the topic  Relations and Functions.

Ordered-Pair Numbers :-

Ordered-pair number is written within a set of parentheses and separated by a comma.

For example, (5, 6) is an ordered-pair number; the order is designated by the first element 5 and the second element 6. The pair (3, 6) is not the same as (6,3) because they have different order. Sets of ordered-pair numbers can represent relations or functions.

Example of ordered pair :

(3,8),(2,1),(7,6)

Relation

A relation is a  set of ordered-pair numbers.

consider the following table

________________________________________________________________________

Numbers of students      1             2          3         4           5          6

_______________________________________________________________________

Marks Obtained             96          98       97         78        77         86

_______________________________________________________________________

In the above table the numbers of students and marks obtained by them  is a relation and can be written as a set of ordered-pair numbers.

A= {(1, 96), (2, 98), (3, 97), (4, 88),(5,77),(6,86)}

When we collect all the elements written in 1st column of the ordered pairs and placed in a set then the set so formed is called  Domain of the relation.
The domain of A= {1, 2, 3, 4,5,6}

As all the elements written in 2nd column of the ordered pairs and placed in a set then the set so formed is called  Range of the relation.

The range of A = {  96,98,97,88,77,86}

we can better understand this concept with the help of this video

Function

A function is a relation in which every first element in ordered pairs have unique second element associated with them. Second  elements may or may not be same.

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Example

 {(1, 2), (2, 3), (3, 4), (4, 5),(5,6)}  is an example of function 

 { (1, 2), (2, 3), (3, 4), (4, 5),(5,6) } is a function because all the  first elements are different.

Example

{(1, 3), (3, 3), (2, 1), (4, 2)}  is an example of function 

 {(1, 3), (2, 3), (2, 1), (4, 2)}  is a function because all the first elements are different.

Example

{ (1, 6), (2, 5), (1, 9), (4, 3) }  is not an  example of function 

As in  {(1, 6), (2, 5), (1, 9), (4, 3)}  the element "1 "   appeared twice .

Example

{(2, 15), (3, 15), (4, 15), (5, 13),(6,18)}  is  an  example of function 

As in  {(2, 15), (3, 15), (4, 15), (5, 15)}   all the first elements are different.

Example

{(1, 1), (-1, 1),(2,4),(-2,4), (3, 9), (-3, 9),(4,16),(-4,16)}  is an  example of function although   the element "1" and "-1" ,"2" and "-2" , "3" and "-3" ,"4","-4" have same images. This is an example of many one function.

Question:-   Find x and y if: 

(i) (5x + 3, y) = (4x + 5,  2)

(ii) (x – y, x + y) = (8, 12)
(iii) ( 2x-y , y+5 ) = ( -2,3 )

Solution

(1)  Given  (5x + 3 , y) = (4x + 5, 2)
So By the equality of ordered pair elements,
1st element of the ordered number written on the left hand side will be equal to the 1st element of the ordered pair number written on the  right hand side . Therefore 

5x + 3 = 4x + 5   and y =  2 
5x-4x = 5 -3   and y = 2 
x = 2 and y = 2

(ii) So By the equality of ordered pair elements
x – y = 8 and  x + y = 12

Solving these two equations for x and y 

 2x =20  and    10+ y =12 

x=10   y = 2

(iii) So By the equality of ordered pair elements
2x-y  =-2  , y+5 = 3 
2x = -2+y  , y = 3-5
2x = -2+y  , y = -2
Putting the value of y in 1st Equation ,we get
2x = -2 - 2
2x = -4
x = -2
so x= -2 and y =-2

Types of Relations

A relation R in a set A is called

(i) reflexive, if (a, a) ∈ R, for every a ∈ A,

(ii) symmetric, if (a, b) ∈ R implies that (a, b) ∈ R, for all a,b ∈ A.

(iii) transitive, if (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R, for all a, b,c ∈ A.

Equivalence Relation

A relation R in a set A is said to be an equivalence  relation if R is reflexive, symmetric and transitive.

1 ) Let B be the set of all triangles in a plane with R a relation in B given by

R = {(T1, T2) : T1 is congruent to T2}. Then R is an equivalence relation.

2 ) Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7}  by

R = {(a, b) : both a and b are either odd or even}. Then R is an equivalence

one-one Function

A function f : X → Y is defined to be one-one (or injection ), if the images of distinct elements of X under f are distinct, i.e., for every x, y ∈ X, f (x) = f (y) implies x = y. Otherwise, f is called many-one.

Onto Function

A function f : X → Y is said to be onto (or surjective), if every element of Y is the image of some element of X under f, i.e., for every y ∈ Y, there exists an

element x in X such that f (x) = y.

Example

1   Function f : R → R, given by f (x) = 2x, is one-one and Onto As all the elements  have only one and uniqe image under f.

2  Function f : N → N, given by f (x) = 2x, is one-one but not onto.Because  the elements  have only one and unique image under f Therefore it is one one function .But not all elements of N have image under f 

e. g .  1,3,5,7... are not the image of any elements of N under f so it is not onto function

Example

The function f : N → N, given by f (1) = f (2) = 1 and f (x) = x – 1,

for every x > 2, is onto but not one-one.

Solution

Since f is Not one-one, as f (1) = f (2) = 1. 

But f is Onto, as given any y ∈ N, y ≠ 1,

Choose x = y + 1 s.t.

 f (y + 1) = y + 1 – 1

f (y + 1)  = y. 

Also for 1 ∈ N, 

we are given  f (1) = 1

Inverse of a Function

A function f : X → Y is defined to be invertible, if there exists a function g : Y → X such that gof = IX and fog = IY. The function g is called the inverse of f and is denoted by f –1

Example

Let S = {1, 2, 3}. Determine whether the functions f : S → S defined as below have inverses. Find f , if it exists.

(a) f = {(1, 1), (2, 2), (3, 3)}

(b) f = {(2, 2), (3, 1), (4, 1)}

(c) f = {(1, 5), (3, 4), (2, 1)}

Solution

(a) It is to  proved that  f is one-one and onto Hence f is invertible with the inverse f –1 of  f given by f –1 = {(1, 1), (2, 2), (3, 3)} = f.

(b) Since f (3) = f (4) = 1, f is not one-one, so that f is not invertible.

(c) Here  f   is one-one and onto, so that f is invertible with

 f –1 = {(5, 1), (4, 3), (1, 2)}.

Composition of Functions

Let f : A → B and g : B → C be two functions. Then the composition of f and g, denoted by gof, is defined as the function gof : A → C given by

gof (x) = g(f (x)), ∀ x ∈ A

Example

fof(x) = (16x + 12 + 18x -12 ) / ( 24x + 18 - 24x +16)

fof(x) = (34 x ) / ( 34)

fof(x) =  x  =  I(x)

Example

Let f : {2, 3, 4, 5} → {3, 4, 5, 9} and g : {3, 4, 5, 9} → {7, 11, 15} be functions defined as f (2) = 3, f (3) = 4, f (4) = f (5) = 5 and g(3) = g(4) = 7 and g(5) = g(9) = 11. Find gof = ?

Solution

We are given

 gof (2) = g (f (2)) 

               = g(3) 

               = 7

 gof (3) = g(f (3)

             = g(4)

              = 7,

gof (4) = g(f (4)) 

           = g(5) 

             = 11 

and  gof (5) = g(f (5))

                   = g (5)                     

                    = 11

So gof ={(2,7),(3,7),(4,11),(5,11)} 

Example

Conclusion

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HOW TO MULTIPLY TWO MATRICES || PRODUCT OF TWO MATRICES

In this post we are  going to discuss  multiplication of two matrices , i.e. when two matrices can be multiplied with each other or when two matrices are NOT eligible for multiplication, what are the necessary conditions for the multiplication of two matrices to be multiplied.

Conditions for Matrix Multiplication

Before multiplication of two matrices we have to check whether multiplication is possible or not , If it is possible then matrices will be multiplied to each other. Necessary condition for multiplication of two matrices is if " The number of columns in the first matrix is the same as the number of rows in the second matrix ". We must know  different types of matrices ,Rows and Column.

Note : The commutation may or may not be possible for multiplication of matrices, That is  in some case AB = BA but In general AB is not equal to BA.

Example 1

1. Multiplication of 2 × 3 matrix with 3 × 4 matrix is possible as number of columns in 1st Matrix is equal to numbers of rows in 2nd Matrix and Resultant matrix will be of 2 × 4 order .

2. Multiplication of 5 × 1 matrix with 1 × 2 matrix is also possible as it gives 5× 2 matrix as resultant Matrix.

3. Multiplication of 4 ×3 matrix with  2 × 3 matrix is NOT possible. Because red colour numbers 3 and 2 do not match .

How to Multiply Matrices 

If condition for matrix multiplication is satisfied, then below is the method of multiplication for two matrices .

Let's take  two matrices A and B  of  2 × 3 and 3×2  orders respectively. The resultant matrix will be  2 × 2 matrix. we start across the 1st row of the first matrix multiplying down the 1st column of the second matrix, element by element. Then we add the resulting products. Our product will be written in position a11 (top left) of the answer matrix which will be equal to   "au+bw+cy".

We shall repeat similar process for the 1st row of the first matrix and the 2nd column of the second matrix. The result will be written  in position a12 which will comes out "av+bx+cz".

Now for the 2nd row of the first matrix and the 1st column of the second matrix. The result will be placed in position a21  as   "du+ew+fy" .Finally, Multiply the 2nd row of the first matrix and the 2nd column of the second matrix to get the product as "dv+ex+fz" The result is placed in position a22. So the resultant matrix will be written as :--

Let us take one  example to Multiply 2×3 and 3×2 Matrices ,The order of resultant matrix will be 2 × 2.

A11 element will be 4×(-3)+1×5+4×6 =  -12+5+24 =17

A12 element will be   4×1+1×6+4×4  =  4+6+16 = 26

A21 element will be  2×(-3)-5×5+7×6 = -6-25+42 = 11

A22 element will be  2×1+(-5)×6+7×4 = 2-30+28 = 0

Since the resultant Matrix  2× 2   as follows

A11  element will be 8×(-2)+9×4      = -16 +36 =20

A12   element will be 8×3+9×0            = 24+ 0 = 24

A21  element will be 5×(-2)+(-1)×4   = -10-4 = -14

A22  element will be 5×3+(-1)×0         = 15+0 = 15
so the resultant matrix will be written as 

To find AB

Since Numbers of columns (3) in 1st matrix (Here A Matrix ) is not equal to numbers of Rows (1) in 2nd Matrix (Here Matrix B ). Therefore matrix multiplication is NOT possible in this case.

To Find BA

Since Numbers of columns (3) in 1st matrix (Here B Matrix ) is equal to numbers of Rows (3) in 2nd Matrix (Here A Matrix ). And resultant Matrix shall be 1×3 order . Therefore matrix multiplication is possible in this case and can be calculated in the following manner :-

BA = ( 4*2+3*3+8*3      4*1+3*(-5)+8*2        4*(-7)+3*0+8*(-1) )

BA = ( 8+9+24    4-15+16    -28+0-8 )

BA = ( 41    5     -36 )

Therefore AB is not equal to BA
This video can better demonstrate the multiplication of two matrices .

Important Note

And what about the matrix multiplication of AB and BA of the matrices given below whose order are 4×3 and 3×3 respectively . 

Then Matrix Multiplication BA is not possible as the numbers of columns(3) in matrix  B is   not  equal to the numbers of rows (4) in  matrix A .

But If we want to Multiply the matrix A with Matrix B as AB then it is possible as the numbers of columns(3) in A matrix is equal to the numbers of rows(3) in B matrix, so it can be calculated as explained earlier. 

It means the order of the matrices play very important role to decide whether matrix multiplication is possible or not. And if possible it is again the order of both the matrices to decide what will be the order of the resultant matrix.

Therefore in  multiplication of matrices of 4×3 and 3×3 .The order of resulting matrix will be 4×3 ,The outer most numbers ( Marked red ). 

Very Important 

If we have two matrices A and B of order 1×3  and 3×1 respectively ,  and if we have find their product AB  and BA , then  

In  1st case  where we have to find AB , let us check its multiplication would be possible or not , as number of columns (3) in 1st matrix is equal to numbers of rows (3) in 2nd Matrix , so AB would be possible and order of resulting matrix would be 1×1 and it can be found using matrix multiplication .

And if we have find their product BA , then would it be possible or not ? let us check .....

In 2nd case where we have to find BA , let us check its multiplication would be possible or not , as number of columns (1)  in 1st matrix is equal to numbers of rows (1) in 2nd Matrix , so AB would be possible and order of resulting matrix would be 3×3 and it can be found using matrix multiplication .

Read this one also  HOW TO UNDERSTAND  BINARY OPERATIONS  ,RELATIONS AND FUNCTIONS   ||  COMMUTATIVE || ASSOCIATIVE

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Conclusion 

This post was about multiplication of two Matrices, Eligibility conditions for product of matrices, Thanks for giving your valuable time to this post . If you liked this post,Please share your valuable opinions about this post. See you in next post ,till then Bye.

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$$\displaystyle{A}={\left(\begin{matrix}-{2}&{1}&{7}\\{3}&-{1}&{0}\\{0}&{2}&-{1}\end{matrix}\right)}    
$$\displaystyle{A}={\left(\begin{matrix}-{2}&{1}&{7}\\{3}&-{1}&{0}\\{0}&{2}&-{1}\end{matrix}\right)}   

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MATRIX , DIFFERENT TYPES OF MATRICES AND DETERMINANTS

Hello every one ,Welcome once again, Today we are going to discuss What is the  matrix, elements of a matrix, order of a matrix, different types of matrix , transpose of matrix, ad joint of matrix, determinant  and  how to find the determinant of a matrix .

What is  Matrix

Matrix definition

A Matrix is a set of elements ( Numbers ) arranged in a particular numbers of Rows and columns in a rectangular table. Matrices inside parentheses ( ) or brackets [ ] is the matrix notation.

Here we have an examples of Matrices . 

The elements which are written in horizontal lines are called Row and elements which are written in Vertical lines are called Column. In the matrix A above ↑ the elements 5, 3, -2 are written in Row wise whereas the elements 5,4,3 are written columns wise, Similarly the elements 4,-1,7 are written in Row wise whereas the elements 3,-1,4 are written columns wise .

Order Of a Matrix

If any matrix have "m" number of rows and "n" number of columns , then "m×n" will be the order of that matrix. It is also called matrix dimensions . For  matrices  given below ,

[      5      6    -4    2    ]   This  matrix has 1×4 order,

    3

[  8 ]  Matrix has 3×1 order ,

   -2 

  [  -1 ] Matrix has 1×1 order.

And the matrices A,B,C and D  above have 3×3 ,2 ×2 , 3×4 and 3×2  respectively, as Matrix A has 3 rows and 3 columns, matrix B has 2 rows and 2 columns, Matrix C has 3 Rows and 4 Columns Similarly Matrix D has 3 rows and 2 columns.

Elements of a matrix

The Elements are entries or numbers used in the matrix . These are denoted by aij , where i is the row's number and j is the column's number in which the element is lying.

Consider the matrix A above which has elements  5 ,3,-2 in the 1st rows 4, -1,7 in the 2nd row and   3,4, -1 in the 3rd rows.

Types of matrix

Equality of Matrices

Two matrices are said to be equal ,if they have same order and same  elements at corresponding positions.

                                                                                                                                                              

If we compare matrices A,B,C and D given above then the matrices A and B may be equal to each other provide x = 6 and y = 3, Similarly Matrices C and D may be equal to each other provided x = 8 ,y = 7 and z = 3 .

Note that Matrices A and C can not be equal to each other, because A and C have not same order. Similarly B and D can not be equal to each other as B and D have different order.

Square Matrix

Any Matrix which have equal numbers of rows and columns,Then   That Matrix is called Square matrix .  A,B,C,D,E and F Matrices given below and above  are the examples of Square Matrix.

Zero , Scalar And Diagonal Matrix

Zero Matrix

If all the elements of a matrix are equal to zero then the matrix  is called Null Matrix ,It is also called Void Or Null Matrix. Matrices A  ,B and C given below are the example of Zero Matrices, As all these matrices have all the elements  zeros. so these matrices are Zero Matrices.

Diagonal Matrix

A Diagonal matrix is a square matrix in which all the elements are zero except diagonal elements . Matrices D ,E and F given below are examples of   Diagonal matrix of  2 × 2 Diagonal matrix and 3×3 orders . 

Scalar Matrix

A Diagonal matrix  matrix in which all the elements are zero and diagonal elements are equal to each others. Matrices A,B,C,D,E and F are examples of Scalar Matrices .Given Matrices D ,E and F given below are examples of   Diagonal matrix of  2 × 2 Diagonal matrix ,3×3  and 3×3 orders respectively . 

Identity Matrix

The Identity Matrix, symbolised as I, is a square matrix. where all the elements are 0 except the diagonal elements  , and all the diagonal  elements equal to one. Identity matrix is also called Unit Matrix.

Above two  examples are  3 × 3 and 2×2 Identity matrices respectively . 1st matrix is an example of  3 × 3 matrix and 2nd is an example of 2×2 Identity matrix. Because both the matrices have diagonals elements one and remaining elements  zeros. Identity Matrices and Unit Matrices are also the examples of Scalar Matrices

Row matrix

Any matrix which has only one row is called Row Matrix.  The  Row matrix may have any numbers of  columns .

[      5      6    -4    2    ] ,     [  2    5   ] ,    [-4    6    -6    0 ]

,[      7      4     3    -3    ]

Column Matrix

Any matrix which has only one column is called Column Matrix. The  Column matrix may have any numbers of Rows.

  

    3

[  8 ] ,      [  -1 ] 

    -2

Transpose of matrix

The transpose of any matrix is obtained by transfer of Rows into Columns and vice versa. i.e transforming 1st row to 1st column and transforming 2nd row to 2nd column and so on for 3rd ,4th and 5th rows and columns.

To find the transpose of any matrix , Shift all the elements of all the Rows into respective Columns. Given below matrices have their transpose written on the right side of them.

It  can be seen that in all the examples given above all the elements of particular row have been changed to corresponding column. So A' is the transpose of  Matrix A ,similarly B' is the transpose of  Matrix B .

Determinant of a matrix

A determinant is a square array of numbers which represents a certain sum of products. we can find out a fixed value of determinant, consider an example of a 3 × 3 determinant , it has 3 rows and 3 columns).

Minor of an Element

The minor of an element aij is the determinant obtained by deleting the ith row and the jth column and is denoted by Mij .

Co factor of an Element

The co-factor of an element aij is the determinant obtained by deleting the ith row and the jth column and is multiplied by(-1)^(i+j)and is denoted by Cij. and Cij= (-1)^(i+j)×Mij

We can find the value of a 2 × 2 determinant   as follows  ,1st we multiply the  top left × bottom right first then subtract from it the product of top right element and left bottom. or 

(1st element in 1st row) × [Its Co factor] - (2nd element in 2nd row)× [Its Co factor].

 

Determinant value of above matrix is 4×5-(-3)×5= 20+15=35

How to find the determinant of a 3x3 matrix  

To find the determinant value 3×3 determinant .

1st element in 1st row ×[Its Co factor]-2nd element in 2nd row[Its Co factor]+3rd element in 3rd row [Its Co factor ].

Let us calculate the determinant value of 3×3 matrix  given above 

=2[(-4×-7) - (2×5)] - (-1) [3×(-7) - (2×5)]+ 5[(3×5) - (-4)×5]

=2[28-10]+1[-21-10]+5[15+20]

=36 - 31+75

=80

Ad joint of a Matrix

The transpose of a co factor matrix of any matrix is called ad joint of the Matrix.To find the ad joint Matrix ,1st find the co factors of  all the elements of given Matrix A.
Co factors of 1st row of  Matrix A are 7 and 5
Co factors of 2nd row of Matrix  A are -2 and -3

,then form the Matrix of these  co factors and name it Co factor  matrix, and after that take the transpose of the co factor matrix so formed.Then we have  transpose of the Ad joint of Matrix.

Conclusion

Thanks for giving your valuable time to the post "What is matrix, element of matrix, dimension of matrix, different types of matrix, transpose of matrix, ad joint of matrix, what is a determinant ,  determinant of 3x3 matrix ,  determinant of a 2x2 matrix " of this blog .If you found this post helpful to you , then share it with yours friends and family members . Also follow me on my blog for notifications of next posts.We shall meet again in next interesting and educating post , till then Good Bye. Take care ....

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