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

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