Matrices Operations-Addition of Matrices

The addition of matrices is defined as two or more matrices. It is quite different from the arithmetic addition of numbers as matrix addition generally follow certain rules. In this article, we will understand in depth the addition of matrices.

However, before we understand the addition of matrices, let us understand what are matrices. A matrix is a rectangular array of numbers that are arranged in rows as well as columns. The horizontal rows are denoted with “m”, whereas the columns are denoted by “n”. So, an order of matrix (m x n) has m number of rows and n number of columns.

Now let us understand

Addition of Matrices

The addition of matrices is one of the elemental operations performed with two or more matrices. The addition of matrices is only possible when the order of the matrices are the same. In this case, we can add the corresponding elements of both matrices. However, matrix addition is not possible if the order is different.

Let us take an example.

A= [aij]m x n and B =[bij] m x n are two matrices so by adding them we get

A+B= [aij]mxn + [bij] m x n = [aij + bij]mxn

Through this, we recall a fundamental concept of the addition of algebraic expressions is that like in algebra. The addition is possible only with the corresponding terms; similarly, the addition of matrices can be done by adding the related terms of the matrix.

Two important criteria define the addition of a matrix. Let us look at them.

1. Let us consider two matrices, A and B. These matrices only can be added when the order of both these matrices are equal, i.e., both of them have the same number of rows and columns. So, for example, if the matrix A is of order 2 x 3 then the matrix B has to be in the order of 2 x 3 to add them.

2. The addition of matrices is not defined for matrices of different sizes.

Properties of Addition of Matrices

The basic properties of the addition of matrices are quite similar to that of real numbers. Understand every property thoroughly as given below

Let us take that A, B and C are three matrices of order m x n, and the following properties are true for adding matrices.

1. Commutative Property

There are two matrices, A and B, and they are of the same order m x n; then, the addition of both these matrices will be commutative, i.e., A + B= B+ A.

2. Associative Property

There are three matrices, A, B and C, and they are of the same order m x n then the addition of three of these matrices are associative, i.e., A + (B+C) = (A+B) + C.

3. Additive Identity

It applies to any m x n matrix with an identity element. So, suppose A is a m x n order matrix, then the additive identity of A will be a zero matrix of the same order in such a way that A + 0 = A (where 0 is the additive identity).

4. Additive Inverse

If A is a matrix of order m x n, then the additive inverse of A would be B (=-A) of the same order, so A+B = 0. So we can say that the sum of the matrix and the additive inverse results would be a zero matrix.

Types of Addition of Matrices

Since you now know the properties of the addition of matrices, let us discuss the types of methods to add matrices. One will be a simple method to add the corresponding elements of two or more matrices. Another is calculating the direct sum of the matrices.

So first, let us understand

Element wise Addition of Matrices

In this type for adding two matrices, we add the elements in each row and column to the corresponding elements of the rows and columns of the next matrix.

For example A= [aij]m x n and B =[bij] m x n, are two matrices so by adding them we get A+B= [aij]mxn + [bij] m x n = [aij + bij] mxn where ij denotes the position of each element in ith row and jth column. 

Direct Sum Matrix Addition

The direct sum of matrices is a less used operation. However, it is important to understand it. Generally, the ⊕ symbol is used while calculating the direct sum of two matrices. In this case, the order of matrices need not be the same.

So, suppose there are two matrices, X and Y, and they are of the order m x n and p x q, respectively, so X would have m rows and n columns and Y would be having p rows and q columns. Then, the matrix X ⊕ Y dimension is (m+p) x (n+q). Also, it is essential to understand that the direct sum of matrices is associative (X⊕Y)⊕Z = X⊕(Y⊕Z). 

Examples on Matrix Addition

1. Addition of Matrices with different order

A=             B=

Here A + B matrix cannot be defined since matrix A is 2 x 2 order and matrix B is 3 x 2. Hence matrices A as well as B can’t be added together. 

2. Addition of Matrices with the same order

Let us take two 3 x 3 matrices

P=     Q=

Hence P+Q can be found by adding elements of P and the corresponding elements of Q.

Value of matrix 

P + Q is

P + Q=

Hence 

P + Q =  

Conclusion

Through this, hopefully, you get an idea about the different types of addition of matrices and the different properties related to the addition of matrices and understand different examples related to the addition of matrices.