Addition of Matrices
A matrix is a rectangular array of numbers, symbols, expressions, letters, etc. arranged in rows and columns. The addition of matrices can be done in different ways but we will mainly discuss the elementwise addition of matrices and the direct sum of matrices in this article. Matrices can be added, subtracted, or multiplied. Here, we will mainly focus on the operation of the addition of matrices.
The addition of matrices is an operation of adding corresponding elements of two or more matrices. The addition of matrices is defined only for matrices of the same size. Let us explore the concept in detail using examples.
1.  What is Addition of Matrices? 
2.  Types of Addition of Matrices 
3.  Addition of 2 × 2 Matrices 
4.  Addition of 3 × 3 Matrices 
5.  Properties of Addition of Matrices 
6.  FAQs on Addition of Matrices 
What is Addition of Matrices?
The addition of matrices is an operation on matrices where corresponding elements of two or more matrices are added. Matrices can be added only if they are of the same size, that is, they have the same dimension or order. A matrix is a rectangular array of numbers, expressions, symbols, etc. arranged in rows and columns. If the number of horizontal rows in a matrix is 'm' and the number of vertical columns is 'n', then the matrix is said to have the dimension 'm × n'.
Addition of Matrices Definition
The addition of matrices is one of the basic operations that is performed on matrices. Two or more matrices of the same order can be added by adding the corresponding elements of the matrices. If A = [\(a_{ij}\)] and B = [\(b_{ij}\)] are two matrices with the same dimension, that is, they have the same number of rows and columns, then the addition of matrices A and B is: A+B = [\(a_{ij}\)] + [\(b_{ij}\)] = [\(a_{ij}+b_{ij}\)]
Types of Addition of Matrices
Now, we will discuss two types of methods to add matrices. One is the simple method to add the corresponding elements of two or more matrices. Another method for the addition of matrices is calculating the direct sum of the matrices. Let us first discuss the former method:
Elementwise Addition of Matrices
The addition of matrices or matrices addition can only be possible if the number of rows and columns of both the matrices are the same. In adding two matrices, we add the elements in each row and column to the respective elements in the row and column of the next matrix. Consider two matrices A and B of the same order 'm × n', where m is the number of rows and n is the number of columns of the two matrices, denoted as, A = [\(a_{ij}\)] and B = [\(b_{ij}\)]. Now, the sum of the two matrices A and B is given as: A+B = [\(a_{ij}\)] + [\(b_{ij}\)] = [\(a_{ij}+b_{ij}\)], where ij denotes the position of each element in i^{th} row and j^{th} column. The dimension of the sum matrix, that is, A + B is also 'm × n'.
Direct Sum  Addition of Matrices
The direct sum of matrices is an operation on matrices that is used less often. The symbol used to denote the direct sum of matrices is ⊕. While calculating the direct sum of two matrices, the order of the matrices need not be the same. Suppose we have two matrices X and Y of orders m × n and p × q, respectively, that is, X has m rows and n columns and Y has p rows and q columns. Then, the dimension of the matrix X⊕Y is (m+p) × (n+q). The direct sum of X and Y, that is, X⊕Y is given in the image below:
As we can see in the image, in the direct sum of matrices, we do not add the corresponding elements of the matrices. The direct sum is a special type of block matrix. The direct sum of matrices is associative, that is, (X⊕Y)⊕Z = X⊕(Y⊕Z).
Addition of 2 × 2 Matrices
Now, let us understand the particular cases of the addition of matrices. Most commonly used matrices have order either 2 × 2 or 3 × 3. Consider two matrices A and B with dimensions 2 × 2. We will add the corresponding elements of the matrices.
Now, let us consider an example of two 2 × 2 matrices A and B and add them.
Addition of 3 × 3 Matrices
Now, we will understand the addition of matrices of order 3 × 3 with the help of an example. As discussed before, for the addition of matrices, the order of the matrices should be the same so that all corresponding elements can be added. The general form to add 3 × 3 matrices is:
Now, let us consider an example of two 3 × 3 matrices A and B and add them.
Please note that for the addition of matrices, matrices need not be square matrices. The addition of rectangular matrices is also defined if the order of the matrices is the same.
Properties of Addition of Matrices
Just like the addition of numbers, the addition of matrices also has similar properties like commutative law, associative law, additive inverse, additive identity, etc. The most important necessity for the addition of matrices to hold all these properties is that the addition of matrices is defined only if the order of the matrices is the same.
 Commutative Property  If A = [\(a_{ij}\)] and B = [\(b_{ij}\)] are two matrices of order m × n, then A + B = B + A.
 Associative Property  If A = [\(a_{ij}\)], B = [\(b_{ij}\)] and C = [\(c_{ij}\)] are three matrices of order m × n, then (A+B)+C = A+(B+C)
 Existence of Additive Identity  If A = [\(a_{ij}\)] is a matrix of order m × n, then the additive identity of A is the zero matrix O of order m × n such that A + O = O + A = A
 Existence of Additive Inverse  If A = [\(a_{ij}\)] is a matrix of order m × n, then the additive inverse of A is A = [\(a_{ij}\)] of order m × n such that A + (A) = O = A + (A)
 Transpose Property  The transpose of the sum of two matrices is equal to the sum of the transposes of the respective matrices, that is, (A + B)^{T} = A^{T} + B^{T}
 Determinant Property  The determinant of the sum of two matrices is equal to the sum of the determinants of the respective matrices, that is, A + B = A + B
Important Notes on Addition of Matrices
 The addition of matrices is defined only if the matrices to be added have the same dimensions.
 Corresponding elements of two or more matrices are added to add the matrices.
 The addition of matrices is closed holds commutative, associative laws and has the additive identity and additive inverse.
Related Topics
Addition of Matrices Examples

Example 1: Write the elements of the sum matrix C = A+B explicitly by addition of matrices A and B of dimension 1 × 2 whose elements are given as: \(a_{11} = 1,a_{12} = 4\) and \(b_{11} = 1, b_{12} = 8\).
Solution: The addition of matrices is defined when the matrices have the same order and we will add the corresponding elements of the two matrices A and B.
Adding the corresponding elements, we have
\(c_{11} = a_{11} + b_{11} \) = 1 + (1) = 0
\(c_{12} = a_{12} + b_{12} \) = 4 + (8) = 4
Answer: \(c_{11} = 0, c_{12} = 4\)

Example 2: Determine the element of second row and third column of the matrix A + B using the addition of matrices definition if \(a_{23}\) = 17 is an element of A and \(b_{23}\) = 20 is an element in B.
Solution: We need to evaluate \(a_{23}+b_{23}\) to determine the element of the second row and third column of the matrix A + B.
\(a_{23}+ b_{23}\) = 17 + 20 = 3
Answer: The element of the second row and third column of A + B is 3
FAQs on Addition of Matrices
What is the Addition of Matrices in Math?
The addition of matrices is the addition of corresponding elements of two or more matrices of the same order. Matrices can be added only if they are of the same size, that is, if they have the same number of rows and columns. If A = [\(a_{ij}\)] and B = [\(b_{ij}\)]. The sum of the two matrices A and B is given as: A+B = [\(a_{ij}\)] + [\(b_{ij}\)] = [\(a_{ij}+b_{ij}\)], where ij denotes the position of each element in i^{th} row and j^{th} column.
How to Do Addition of Matrices?
Corresponding elements of matrices A and B of the same dimensions are added to determine the sum matrix A + B. If A = [\(a_{ij}\)] and B = [\(b_{ij}\)], the sum of the two matrices A and B is given as: A+B = [\(a_{ij}\)] + [\(b_{ij}\)] = [\(a_{ij}+b_{ij}\)], where ij denotes the position of each element in i^{th} row and j^{th} column.
What is the Necessary Condition for Addition of Matrices?
For the addition of matrices, the necessary condition is for them to have the same number of rows and columns. Matrices to be added should have the same dimension.
Is Addition of Matrices Commutative?
Addition of matrices is commutative as when two matrices A = [\(a_{ij}\)] and B = [\(b_{ij}\)] of same order are added, then A + B = [\(a_{ij}+b_{ij}\)] = [\(b_{ij}+a_{ij}\)] = B + A.
Is the Addition of Matrices possible for Matrices of Different Dimensions?
No, the addition of matrices is possible only if the matrices to be added have the same dimensions. When two or more matrices are added, we add all the corresponding elements of the matrices but if the order is different then adding all corresponding elements is not possible.