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 = [a_{ij}]_{m × n }and B = [b_{ij}]_{m × n}
Let A + B = C = [c_{ij}]_{m × n} and B + A = D = [d_{ij}]_{m × n}
Then, c_{ij }= a_{ij} + b_{ij }
= b_{ij} + a_{ij }, (by using the definition of addition of matrices)
= d_{ij }
Since C and D are of the same order and c_{ij }= d_{ij }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 = [a_{ij}]_{m × n },B = [b_{ij}]_{m × n} and C = [c_{ij}]_{m × n}
Let B + C = D = [d_{ij}]_{m × n} , A + B = E = [e_{ij}]_{m × n} , A + D = P = [p_{ij}]_{m × n} , E + C = Q = [q_{ij}]_{m × n }
Then, d_{ij }= b_{ij} + c_{ij }, e_{ij }= a_{ij} + b_{ij }, p_{ij }= a_{ij} + d_{ij} and q_{ij }= e_{ij} + c_{ij}
Now, A + (B + C) = A + D = P = [p_{ij}]_{m × n}
and (A + B) + C = E + C = Q = [q_{ij}]_{m × n}
Therefore, P and Q are the matrices of the same order and
p_{ij} = a_{ij }+ d_{ij }= a_{ij} + (b_{ij }+ c_{ij})
= (a_{ij} + b_{ij})_{ }+ c_{ij} , (by the definition of addition of matrices)
= e_{ij }+ c_{ij}
= q_{ij}
Since P and Q are of the same order and p_{ij }= q_{ij }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 = [a_{ij}]_{m × n }and O = [0]_{m × n}
Therefore, A + O = [a_{ij}] + [0]
= [a_{ij} + 0]
= [a_{ij}]
= A
Again, O + A = [0] + [a_{ij}]
= [0 + a_{ij}]
= [a_{ij}]
= 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 = [a_{ij}]_{m × n}
Therefore,  A = [ a_{ij}]_{m × n}
Now, A + ( A) = [a_{ij}] + [ a_{ij}]
= [a_{ij} + ( a_{ij})]
= [0]
= O
Again ( A) + A = [ a_{ij}] + [a_{ij}]
= [(a_{ij}) + a_{ij}]
= [0]
= O
Therefore, A + ( A) = O = ( A) + A
Note: The matrix – A is called the additive inverse of the matrix A.
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