Addition of Matrices

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.

Addition of Matrices

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]

Types of Addition of Matrices

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:
   1. Add a and b
   2. Add m and n
   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)

Properties of Addition of matrices

• Transposition: If the sum obtained by adding two A and B matrices is transposed, it will be equal to the sum obtained by adding the transposes of the two matrices A and B.
• Additive identity: If a zero matrix O is added to a matrix A whose dimensions are mxn, then the result will be A. So, A+O = O+A = A. Hence, the zero matrix is an additive identity in the addition of matrices.
• Commutative property: If there are two matrices A and B of the same order, then A+B = B+A. This is the commutative property in the addition of matrices.
• Associative property: The addition of matrices is associative because if there are three matrices A, B, and C of the same dimensions, then (A+B)+C = A+(B+C).
• Additive inverse: If A is a matrix of the order jxk and -A is a matrix of the same or jxk where all its elements are the same as the matrix A but have a sign that is opposite to the corresponding elements of A, then A + (-A) = O = A + (-A). And -A is an additive inverse.
• Determinant property: If the determinants of two matrices are added, the sum is the determinant of the sum of the original matrices.

Conclusion

The addition of matrices can be done only if the dimensions of the matrices are the same. If a direct sum of the matrices is found, then the result is a special kind of block matrix. That is why, the direct method of finding the sum of matrices is rarely used. In the element-wise addition of matrices, the corresponding elements of the matrices are added. That is why the addition of matrices displays so many properties of the mathematical operation of addition.