# Addition of Matrices – Properties and Examples

Here you will learn how to add matrix and properties of addition of matrices with examples.

Let’s begin –

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

Thus, if A = $$[a_{ij}]_{m\times n}$$ and B = $$[b_{ij}]_{m\times n}$$ are two matrices of the same order, their sum A + B is defined to be the matrix of order $$m\times n$$ 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 by only when they are of the same order.

Example : If A = $$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$$, B = $$\begin{bmatrix} 6 & 5 & 4 \\ 3 & 2 & 1 \end{bmatrix}$$, then

A + B = $$\begin{bmatrix} 1 + 6 & 2 + 5 & 3 + 4 \\ 4 + 3 & 5 + 2 & 6 + 1 \end{bmatrix}$$ = $$\begin{bmatrix} 7 & 7 & 7 \\ 7 & 7 & 7 \end{bmatrix}$$

If A = $$\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$$, B = $$\begin{bmatrix} -1 & 2 & 1 \\ 3 & 2 & 1 \\ 2 & 5 & -2 \end{bmatrix}$$, then A + B is not defined, because A and B are not of the same order.

(a) Commutativity :  If A and B are two $$m\times n$$ matrices, then A + B = B + A. i.e. matrix addition is commutative.

(b) Associativity : If A, B, C are three matrices of the same order, then (A + B) + C = A + (B + C) i.e. matrix addition is associative.

(c) Existence of Identity : The null matrix is the identity element for matrix addition.

(d) Existence of Inverse : for every matrix A = $$[a_{ij}]_{m\times n}$$ there exist a matrix $$[-a_{ij}]_{m\times n}$$, denoted by -A, such that A + (-A) = O = (-A) + A

(e) Cancellation Laws : If A, B, C are matrices of the same order, then

A + B = A + C $$\implies$$ B = C

and, B + A = C + A $$\implies$$ B = C