Addition of Matrices
2026-02-28 10:06 Diff
https://brightchamps.com/en-us/math/algebra/addition-of-matrices

The addition of a matrix is an operation where the corresponding elements of matrices of the same dimensions are added together.

The addition of a matrix is only possible when the matrices have the same dimensions. In a matrix, the number of horizontal rows is denoted by m, and the number of vertical columns by n; the matrix is said to have the dimension m × n.

Types of Addition of Matrices


In addition, matrices are categorized into two types, which depend on the type of matrix and the properties used to solve the problems. Here are two types of addition methods: the simple method and the direct sum of matrices. Let us see in detail.

Element-Wise Addition of Matrices

Element-wise addition of a matrix is the basic form of matrix addition. This is otherwise known as the simple method of matrix addition.

This method is used in a matrix, where corresponding elements from the matrices of the same order are added together. This means that elements are in the same rows, and the columns are added to form a new matrix.

For example:

\(A = \begin{bmatrix} 2 & 6 \\ 4 & 8 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 5 \\ 3 & 7 \end{bmatrix}\)

In this matrix, they are in 2 × 2 the same order.

Add the elements step by step:

\(A + B = \begin{bmatrix} 2 & 6 \\ 4 & 8 \end{bmatrix} + \begin{bmatrix} 1 & 5 \\ 3 & 7 \end{bmatrix} = \begin{bmatrix} 2+1 & 6+5 \\ 4+3 & 8+7 \end{bmatrix}\)

The final result is:

A + B \(= \begin{bmatrix} 3 & 11 \\ 7 & 15 \end{bmatrix}\)

Direct sum matrices

The direct sum method is a method that is used in matrices to add two or more matrices. When adding the matrices in the direct sum matrix, the order of the matrices is not the same.

In a direct sum matrix, which is denoted by ⊕. The direct sum is not like normal addition. A direct sum matrix means combining two small matrices into a bigger one, placing each one along the diagonal and filling the other positions with zeros.

For example: 
\(A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 7 & 9 \\ 6 & 8 & 10 \end{bmatrix}\)

Solution:

\(A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}\), which is 2 × 2 matrix \( B = \begin{bmatrix} 5 & 7 & 9 \\ 6 & 8 & 10 \end{bmatrix}\), which is a 2 × 3 matrix. These two matrices are not the same. Therefore, we will use a direct sum matrix. A ⊕ B.

While using this method, keep the first matrix on the top left side and keep the second matrix on the bottom right side, then add the zero in between the spaces.

\(A \oplus B = \begin{bmatrix} 1 & 3 & 0 & 0 & 0 \\ 2 & 4 & 0 & 0 & 0 \\ 0 & 0 & 5 & 7 & 9 \\ 0 & 0 & 6 & 8 & 10 \end{bmatrix}\)