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The addition of matrices is the mathematical process of adding two or more matrices. Matrices are a rectangular arrangement of numbers, letters, symbols, expressions, which form rows and columns in a rectangular matrix. The mathematical operations of addition, subtraction, multiplication, and division can be carried out on matrices.

Matrices allow computations of linear algebra, and since most properties and calculations of linear algebra are applicable on matrices, matrices form a large part of linear algebra. This is an example of a matrix:

The above matrix has two rows and three columns. It is said to be a 2×3 matrix.

The addition of matrices, specifically, means the addition of the corresponding elements of the matrices under advisement. The rules for the addition of matrices have been set down only for matrices that are of the same dimension, which means only matrices that have the same number of rows and columns can be added together.

The addition of matrices is one of the most basic operations performed on matrices. Two or more matrices can be added by adding their corresponding elements. This means that if two or more matrices have the same number of rows and columns, they are eligible to be added together. So, for example, if A = [a] and B = [b] then A+B = [a]+[b] = [a+b]

There are mainly two ways used in the addition of matrices. One is the element-wise or simple addition where the corresponding elements are added, and the other method is finding out the direct sum of the matrices. Let us consider the former method of addition:

1. Simple addition of matrices: The simple way of adding matrices is to simply add the corresponding elements of the matrices that need to be added. For this to be possible, the matrices need to have the same number of rows and columns. In this type of addition of matrices, the element belonging to a particular row and column is added to the element of the other matrix or matrices of the same row and column. So, for example, if there are two matrices, A and B, that need to be added, an element ‘a’ of the matrix A, which is situated in row 1 and column 2, will be added to the element of B, which is situated in row 1 and column 2.
2. The direct sum method: Though rarely used, this method can determine the sum of matrices of different dimensions. The symbol ⊕ is used to denote the sum of two matrices calculated by the direct method. So, if there are two matrices, D and F of the order axb and mxn, where D has a rows and b columns, and F has m rows and n columns. Then, in order to determine the dimensions of D⊕F, the following procedure needs to be followed:
3. Multiply the two sums obtained
4. The equation would be something like this (a+b)x(m+n)

So in the direct sum of the matrices, the corresponding elements are not added. So the direct sum of matrices is a block matrix. A property of the direct sum of matrices is that it is associative, that is, (A⊕B)⊕C = A⊕(B⊕C)