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.