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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