Properties of Addition of Matrices | Commutative Law

0
10


We will discuss about the properties of
addition of matrices.

1. Commutative Law of Addition of Matrix:
Matrix multiplication is commutative. This says that, if A and B are matrices
of the same order such that A + B is defined then A + B = B + A.

Proof:  Let A = [aij]m × n and B
= [bij]m × n

Let A + B = C = [cij]m × n
and B + A = D = [dij]m × n

Then, cij = aij + bij

              = bij + aij ,
(by using the definition of addition of matrices)

              = dij

Since C and D are of the same order and cij
= dij then, C = D.

i.e., A + B = B + A. This completes the
proof.


2. Associative Law of Addition of Matrix:
Matrix addition is associative. This says that, if A, B and C are Three
matrices of the same order such that the matrices B + C, A + (B + C), A + B, (A
+ B) + C are defined then A + (B + C) = (A + B) + C.

Proof: Let A = [aij]m × n ,B
= [bij]m × n and C = [cij]m × n

Let B + C = D = [dij]m × n
, A + B = E = [eij]m × n , A + D = P = [pij]m
× n
, E + C = Q = [qij]m × n

Then, dij = bij + cij
, eij = aij + bij , pij = aij
+ dij and qij = eij + cij

Now, A + (B + C) = A + D = P = [pij]m
× n

and (A + B) + C = E + C = Q = [qij]m
× n

Therefore, P and Q are the matrices of the
same order and

              pij = aij + dij = aij
+ (bij + cij)

                   = (aij
+ bij)+ cij , (by the definition of addition
of matrices)

                    = eij + cij

                    = qij

Since P and Q are of the same order and pij
= qij then, P = Q.

i.e., A + (B + C) = (A + B) + C. This
completes the proof.

3. Existence of Additive Identity of
Matrix:
Let A be the matrix then, A + O = A = O + A

Therefore, ‘O’ is the null matrix of the
same order as the matrix A

Proof: Let A = [aij]m × n and
O = [0]m × n

Therefore, A + O = [aij] + [0]

                          = [aij
+ 0]

                          = [aij]

                           = A

Again, O + A = [0] + [aij]

                     = [0 + aij]

                     = [aij]

                     = A

Note: The null matrix is called the
additive identity for the matrices.

4. Existence of Additive Inverse of Matrix:
Let A be the matrix then, A + (- A) = O = (- A) + A

Proof: Let A = [aij]m × n

Therefore, – A = [- aij]m ×
n

Now, A + (- A) = [aij] + [- aij]

                       = [aij + (-
aij)]

                       = [0]

                        = O

Again (- A) + A = [- aij] + [aij]

                       = [(-aij) +
aij]

                       = [0]

                       = O

Therefore, A + (- A) = O = (- A) + A

Note: The matrix – A is called the additive
inverse of the matrix A.

`

10th Grade Math

From Properties of Addition of Matrices to HOME


Didn’t find what you were looking for? Or want to know more information
about
Math Only Math.
Use this Google Search to find what you need.








Source link

LEAVE A REPLY

Please enter your comment!
Please enter your name here