`text(Addition)`

Let A, B be two matrices, each of order `m xx n` . Then, their sum `A + B` is a
matrix of order m x n and is obtained by adding the corresponding elements of A
and B.
Thus, if `A= [a_(ij)] m xx n` and `B= [b_(ij)], m xx n` then `A + B= [a_(ij) + b_(ij)]m xx n AA i, j`

`Ex-` `A= [ (1,3,5),(-2,0,2),(0,4,-3)] , B = [(0,3),(-2,0),(0,-4)]` and `C = [(4,1,-2),(3,2,1),(2,-1,7)]`

Find (whichever defined)
(i) A+ B (ii) A+ C

(i) Given, A is a matrix of the type `3 xx 3`
and B is a matrix of the type `3 xx 2.`
Since, A and B are not of the same type.
`therefore` Sum A + B is not defined.
(ii) As A and C are two matrices of the same type, therefore the sum A + C
is defined.

`therefore` `A+ C = [ (1,3,5),(-2,0,2),(0,4,-3)] + [(4,1,-2),(3,2,1),(2,-1,7)]`
`= [(1+4,3+1,5-2),(-1+3,0+2,2+1),(0+2,4-1,-3+7)] = [(5,4,3),(1,2,3),(2,3,4)]`



`text(Properties of Matrix Addition :)`

`text(Property 1)` Addition of matrices is commutative
i.e., A+B=B+A
where A and Bare any two m x n matrices i.e., matrices of the same order.

`text(Property 2)` Addition of matrices is associative
i.e., (A+ B)+ C = A+ (B +C)
where A, Band Care any three matrices of the same order m x n (say).

`text(Property 3)` Existence of additive identity
i.e., A+ o = A= o + A
where A be any m x n matrix and 0 be the m x n null matrix. The null
matrix 0 is the identity element for matrix addition.

`text(Property 4)` Existence of additive inverse
If A be any m x n matrix, then there exists another m x n matrix B, such
that A+B=O=B+A
where O is the `m xx n` null matrix.
Here, the matrix B is called the additive inverse of the matrix A or the
negative of A.

`text(Property 5)` Cancellation laws
If A, Band Care matrices of the same order `m xx n` (say), then
`A+ B = A+ C => B = C` [left cancellation law]
and `B + A = C + A => B = C` [right cancellation law]