MATRIX , DIFFERENT TYPES OF MATRICES AND DETERMINANTS

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,

 

  [  -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. 

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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HOW TO FIND THE TRANSPOSE OF MATRIX WITH AN EASY METHOD

What is the Transpose of these Matrices ??

What is transpose of matrix, Method to find inverse of a matrix ,Matrices and Determinants in Most imp, How to find inverse of 3x3 Matrix, how to solve system of equation,matrix method of solving system of equations of three variables,determinant of 3x3 matrix calculator,

As it is clear from its name  transpose means trans + pose i. e. transfer + position ( transfer of position ) ,  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. This is very easy and interesting topic in matrices and determinants.How to find the Transpose of Matrix ,

 To find the transpose of the matrix 

   

1^st of all  shift all the elements which are in 1st row to 1^st column as

5

5

2
,then shift the elements which are in      2nd row to 2^nd column as  

-1

-3

 7

similarly  shift all the elements which are in 3rd row to 3^rd column as

4

2

8

And the matrix so obtained is the transpose matrix. We can  check that same colour row have been  transformed to same colour column
        

Now we shall take one  example to find the transpose matrix

Here Given matrix have 3 rows and 4 columns .It means we shall have 4 rows and 3  columns  in transpose matrix.

1st shift all the elements which are in 1st row to 1^st column as

5
8               
6
-4

shift all the elements which are in 2nd row to 2nd column as

3
5
8
3

shift all the elements which are in 3rd row to 3rd column as 

-3
8
-7
6

so shifting the corresponding Rows into Corresponding columns. We can  check that same colour rows are transformed to same colour columns.

Now we shall take one more  example to find the transpose of matrix .Here Given matrix have 3 rows and 4 columns,It means  we shall have 4 rows and  3 columns in transpose of that matrix.

1st shift all the elements which are in 1st row to 1^st column as

 7

-1

-2

 5

shift all the elements which are in 2nd row to 2nd column as

4

4

3

3

shift all the elements which are in 3rd row to 3rd column as 

  8

 9

 6

-1

so shifting the corresponding Rows into Corresponding columns. We can that check same colour rows are transformed to same colour column.

Let us take an example  Where  A =

Step 1 Then  on transforming 1st Column  to 1st Row  ,we have 

4             -3         9          as 1st  Row

Step 2 Then  on transforming  2nd Column  to 2nd Row  ,we have 

5             2          -2          as  2nd Row

||ly    on transforming   3rd  Column  to 3rd Row  ,we have 

7          3          8             as 3rd Row

After  taking Transpose  A' will be  

Let us take more example to find out the transpose of matrices. These are two examples , both of which are of  3×3 orders. Hence the transpose of these matrices will be  again 3×3. 


1st consider matrix G , after transforming its 1st row into column , the 1st column of the transpose matrix of  G '  will be  -2,  -5,  4.
After transforming its 2nd row into column , the 2nd column of the transpose matrix of  G '  will be   -5,  7 ,  3. And after transforming its 3rd row into column , the 3rd  column of the transpose matrix of  G '  will be  4,  3,  8. If we write the transpose of  matrix G then we can see that there is no difference between the matrix G and the transpose of the matrix G.

Now consider matrix H , after transforming its 1st row into column , the 1st column of the transpose matrix of  G '  will be  2,  3,  4.      
After transforming its 2nd row into column , the 2nd column of the transpose matrix of  H '  will be   3, 5 ,  6. And after transforming its 3rd row into column , the 3rd  column of the transpose matrix of  H '  will be  4,  6,  7. Again in the case of matrix H , Matrix H and its Transpose matrix   H '  are same.

Here is more interesting Example of Transpose of this Matrix,This matrix have 3 Rows and 3 columns,after taking Transpose this matrix still have 3 Rows and 3 columns,

Now Take Transpose of this Matrix 

what ?

Surprise to see that the Transpose of some of the Matrices are the Matrices Itself, i. e. if A¹= A

Such Matrices are called Symmetric  Matrices.

Again if we take the transpose of the matrix given below and take -1 common from the matrix so obtained i .e  this matrix will be equal to transpose of the negative of the transpose of the given matrix

then   A¹ =  - A ,  Such Matrices are called  Skew Matrices.

Final words

This post was regarding what is transpose of matrix,finding the transpose of the matrix in matrices and determinants,Method to find inverse of a matrix ,Matrices and Determinants in Most imp, How to find inverse of 3x3 Matrix If you learn something from this post then share it with your friends and also follow me on my blog ,We shall meet again in next post , till then Good Bye ...........................

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