EQUALITY OF MATRICES

Definition

Two matrices A = [ a_ij ]_m×n and B = [ b_ij ]_r×s are equal if

1. m = r , that is , the number of rows in A  equals the number of rows in B.
2. n = s , that is , the number of columns in A equals the number of columns in B.
3. a_ij = b_ij for i = 1,2,3,....,m and j = 1,2,3,...,n.

 

If two matrices A and B are equal, we write

       A = B ,otherwise we write A ≠B.

For example --

The matrices A = 

B = 

are equal if x = -1 , y = 0 and z = 4.

Illustrative Examples

(1)A matrix has 12 elements. What are the possible orders it can have?

Sol.

We know that if a matrix is of order m×n, then it has mn elements. Therefore, to find all possible orders of a matrix with 12 elements, we will have to find all ordered pairs (a,b) such that a and b are factors of 12.

Thus , all ordered pairs of this type are

    (1,12),(12,1),(3,4),(4,3),(2,6),(6,2)

Hence , possible orders of the matrix are

    1×12,12×1,3×4,4×3,2×6,6×2

(2)Construct a 2×3 matrix A=[a_ij] whose elements are given by a_ij = (i -j)/(i +j).

Sol.

We have a_ij = (i -j)/(i +j)

Therefore , a₁₁ = (1-1)/(1+1)

                        =0

   a₁₂ = -1/3 ,a₁₃= -½ , a₂₁ =1/3 ,a₂₂=0 and

   a₂₃ = -1/5

So the required matrix is A =

(3)If A =[a_ij] is a matrix given by

       A =[a_ij] =

Write the order of A and find the elements a₂₃ , a₃₁ .

Sol.

We observe that there are 3 rows and 3 columns in a matrix A .Therefore , it is of order 3×3 .

And , a₂₃ = 9 and a₃₁ =21 .

(4)If      =, find the value of a and b .

Sol.

Since the corresponding elements of two equal matrices are equal .

Therefore ,  =

Thus , a +b = 6 and ab = 8

Solving these two equations , we get ,

           a =2 , b =4 or a =4 , b =2.

(5)If A = 

then , find (i)a₂₂ +a₂₃

                 (ii )a₁₁ +a₂₂ +a₃₃

Sol.

Given A = 

Then ,

(i)a₂₂ +a₂₃ = 4+9=13

(i i)a₁₁ +a₂₂ +a₃₃ =2+4+(-2) =4

ADDITION OF MATRICES

Definition

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

Thus , if A =[a_ij ]_m×n and B =[b_ij ]_m×n are two matrices of the same order , their sum A+ B is defined to be the matrix of order m×n such that

        ( A +B)ij = a_ij + b_ij  for i = 1,2,...,m and j = 1,2,....,n

Note

The sum of two matrices is defined only when they are of the same order .

For example -

If A =  , B =  , then

A + B =  + 

        

          = 

      

          = 

PROPERTIES OF MATRIX ADDITION

(1)Commutativity

If A and B are two m×n matrices , then

         A + B = B + A

That is , matrix addition is commutative .

(2)Associativity

If A ,B ,C are three matrices of the same order , then

 (A +B ) + C = A + ( B + C )

That is, matrix addition is associative .

(3)Existence of Identity

The null matrix is the identity element for matrix addition , that is

          A + O = O +A

(4)Existence of Inverse

For every matrix A = [a_ij ]_m×n there exists a matrix [ -a_ij ]_m×n , denoted by - A , such that

       A + (- A) = O = (- A) + A

(5)Cancellation Laws

If A , B ,C are matrices of the same order , then

   A + B = A + C  ⇒B = C   ( Left cancellation law )

and ,    B + A = C + A  ⇒B = C   ( Right cancellation law )