3.5 Types of Matrices:

 

Definitions:

‘Square Matrix’ is a matrix whose number of rows are same as number of columns. Square matrix’s order is represented by m (m x m)

A=  order :3X3

B=  order: 2X2

 

 

The Principal diagonal elements (from top left corner to bottom right corner) of matrix A are {1,5,9}

 

 

The Principal diagonal elements (from top left corner to bottom right corner) of matrix A are {1,4}

 

A= order :3X2

B= order :2X3

 

Note Matrix A is  3X2   and Matrix B is 2X3

 

Since these two are not square matrices, we cannot identify their diagonal elements.

 

A matrix whose principal diagonal elements are non zero and all other elements are zero is called ‘Diagonal matrix.’

 

A=Principal diagonal elements are {2,4,6}

B= Principal diagonal elements are {2,4,6}

 

 

 

Observe that in both A and B except for diagonal elements all other elements are zero.

 

 

A ‘Scalar matrix’  is a diagonal matrix whose principal diagonal elements are equal

A= Principal diagonal elements are {2,2,2}

 B= Principal diagonal elements are {5,5}

 

 

 

An ‘Identity matrix’  is a diagonal matrix whose principal diagonal elements are 1.

A=Principal diagonal elements are {1,1,1}

B= Principal diagonal elements are {1,1}

 

 

 

‘Symmetric matrix’ is a square matrix whose elements are symmetric (same) with respect to principal diagonal elements.

(Mirror copy with respect to principal diagonal).

A=Principal diagonal elements are {5, 9,6}

B= Principal diagonal elements are {7, 9}

 

 

Notice that in A the elements on both the sides of principal diagonal are same  

{-2,-2},{-4,-4},{6,6}.

 

 

Notice that in B the elements on both the sides of principal diagonal are same   {-2,-2}.

Skew symmetric matrix’ is a square matrix whose elements are symmetric with respect to the principal diagonal with opposite sign and principal diagonal elements are zero

A=Principal diagonal elements are {0,0,0}

 

B= Principal diagonal elements are {0,0}

 

 

Notice that in A the elements on both the sides of principal diagonal are same with opposite sign

{-2,2},{4,-4},{-6,6}.)

 

Notice that in B the elements on both the sides of principal diagonal are same with opposite sign

{-2,2}.)

A= order 1X4

B= order 1X2

Row matrix’ is a matrix which has only one row and is of the order (1 x n)

 

A= order: 4X1

B= order :  2X1

 

 

 

 

 

‘Column  matrix’ is a matrix which has only one column  and is of order (m x 1)

A=  order 3X4 

B=  order 2X3

 

 

 

 

‘Zero  matrix’ is a matrix whose elements are all zeros.

 

It need not be a square matrix

 

 

A=

B=    Then A=B

 

 

 

 

Two matrices are said to be ‘equal’ if and only if they are of same order and corresponding elements are equal.

A=                                      

 

B =  If A=B, then

 

 

 

 

 

 

 

 

a=1, b=2,c=3,d=4,e=5,f=6,g=7,h=8,i=9,j=2,k=4,l=6.  Rows  Columns

 

‘Transpose of a matrix’   is obtained by converting elements of rows in to columns and elements of columns in to rows.

Transpose of A is denoted by A1.

A=   order   4X3

 

A1=  order   3X4

 

 

Rows:  {2,4,6},{8,9,1},{3,5,7},{2,4,6}. Columns:  {2,8,3,2},{4,9,5,4},{6,1,7,6}

 

 

 

Rows:  {2,8,3,2},{4,9,5,4},{6,1,7,6}. Columns:  {2,4,6},{8,9,1},{3,5,7},{2,4,6}

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                                                                    

 

3.5 Summary of learning

 

 

No

Points studied

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2

Types of matrices - Square matrix, rectangular matrix, diagonal matrix, Symmetric matrix, Zero matrix, Skew matrix, Identity matrix

Transpose of a matrix