How to Describe a Elementary Matrices

In order to locate the position of a particular element of a matrix. We have to specify  the number of the row and that of  the column in which the element occurs. An element occurring in the ith row and jth column of a matrix A will be called the (i,j)th element of A , to be denoted by aij .

In general ,an m x n matrix A may be written as ,

A  =[[a11,a21,a1n],[a21,a22,a2n],[...,...,...],[am1,am2,amn]]



Operation on matrices

Basic operation involved in matrices are,

1)Addition of matrices

2)Subtraction of matrices

3)Multiplication of matrices

Let  us discuss about addition of matrices.

Addition of Matrices:

Let A and B be two comparable matrices,each of order (m xn).Then their sum(A+B) is a  matrix of order (m x n) ,obtained by adding the corresponding elements of A  and B.

Thus, ifA= [aij]mxn    and  B = [bij] mxn   then

A+B = [aij  +bij] m xn

Note : for two matrices A and B ,the sum (A+B) exits only when A and B are comparable.

Examples of Adding matrix:

Example 1:

if A= [[3,4],[5,7]] and B  =[[4,6,1],[5,7,3]] then A and B are matrices of order 2 x2 and 2 x3 respectively.

So A and B are not Comparable.

Hence ,A + B is not defined.

Example 2:

Let A=[[3,6,9],[3,5,1]] and B=[[-4,8,5],[8,3,-2]]

Clearly ,each one  of A and B is a 2 x 3 matrix.

So a and B are comparable matrices.

Therefore A+B is defined.

Now,   A+ B  =  [[3 + (-4) ,6+8 , 9+5],[3+8 , 5+3 , 1+(-2)]]


=[[-1,14,14],[11,8,-1]]

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Some Resultes on Addition of matrices.

Theorem:

Matrix addition is commutative. ie)A+B = B+A for all comparable matrices A and B

PROOF:

Let A =[aij]mxn   and B = [bij]mxn Then,

A+B= [aij]mxn  +[bij]mxn

=[aij +bij]mxn [by the definition of addition of matrices]

=[bij+aij]mxn [Addition of numbers is Commutative]

=[bij]mxn +[aij]mxn  =B+A

Hence ,A+b = B+A

Check  the theorem:

Let A =[[2,4],[5,4]] and B=[[4,3],[2,5]]

A+B  =[[2+4,4+3],[5+2,4+5]]

=[[6,7],[7,9]]

B+A =[[4+2,3+4],[2+5,5+4]]

=[[6,7],[7,9]]

A+B = B+A

Hence proved.

Theorem 2:

Matrix Addition  is Associative ie )(A+B+ +C  =A+(B+C)

PROOF: Let A=[aij]mxn, B=[bij]mxn and C=[cij]mxn Then

(A+B) +C=([aij]mxn +[bij]mxn+[cij]mxn)

=[aij+bij]mxn +[cij]mxn

=[(aij+bij) +[cij]mxn

=[aij +(bij +cij)]mxn  [addition of numbers is associative]

=[aij]mxn +[bij +cij]mxn

=[aij]mxn +([bij]+cij])mxn)  =  A+(B+C)

Hence ,(A+B)  +C = A+(B+C)

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Check the theorem 2:

Let A=[[2,4],[5,4]] ,B=[[3,4],[5,3]] and C=[[8,4],[3,5]]

Prove that (A+B)  +C  = A+(B+C)

Take (A+B) +C  => [[2+3,4+4],[5+5,4+3]] +[[8,4],[3,5]]

=> [[5,8],[10,7]]   + [[8,4],[3,5]]

=> [[5+8,8+4],[10+3,7+5]]

(A+B) +C   => [[13,12],[13,12]]

Take A+(B+C) => [[2,4],[5,4]]   +  [[3+8,4+4],[5+3,3+5]]

=> [[2,4],[5,4]]    +  [[11,8],[8,8]]

=> [[2+11,4+8],[5+8,4+8]]

A+(B+C)             => [[13,12],[13,12]]

Therefore ,(A+B) +C =A+(B+C)

Hence Proved.

Theorem 3:

if A is an mxn matrix and O=[bij]mxn

where bij =0 for all suffixes i andj

Then,   A+O = [aij]mxn +[bij]mxn  =[aij +bij]mxn

=[aij +0]mxn  [bij  =0]

=[aij] mxn  =A

A+O=A

Similarly,O+A =A

Hence A+O=O+A =A

Check the theorem 3

Let A =[[3,5,4],[2,7,5]] and O=[[0,0,0],[0,0,0]]

A+O  =[[3+0,5+0,4+0],[2+0,7+0,5+0]]

A+O  =[[3,5,4],[2,7,5]]

And O+A  =[[0+3,0+5,0+4],[0+2,0+7,0+5]]

=[[3,5,4],[2,7,5]]

Therefore ,A+O  =O+A

Hence Proved.