Matrix Addition is used to find the sum of two matrices of same size. In this class, we will learn how to do matrix addition of given matrices, with the help of examples.

What is Matrix Addition?

Matrix addition is the process of adding two matrices by adding the corresponding entries together.

\({\ \begin{bmatrix}a_{11} & a_{12}\\a_{21} & a_{22}\end{bmatrix} }\) + \({\ \begin{bmatrix}b_{11} & b_{12}\\b_{21} & b_{22}\end{bmatrix} }\) = \({\ \begin{bmatrix}a_{11} + b_{11} & a_{12} + b_{12}\\a_{21} + b_{21} & a_{22} + b_{22}\end{bmatrix} }\)

Requirements for Matrix Addition

The operation is defined only for matrices of the same dimensions, meaning that each matrix must have the same number of rows and columns as the other.

If Matrix A is of dimension m x n, Matrix B must also be of dimension m x n.

How to do Matrix Addition

To perform matrix addition, follow these steps:

1. Ensure that the matrices you are given to add have the same dimensions. If so, then proceed with the next step. Else, you may say that the Matrix addition is not valid.

2. Add the corresponding elements from each matrix to form a new matrix.

Addition of 2x2 Matrices

Consider two matrices A and B of dimension 2x2:

Matrix \({\ A = \begin{bmatrix}a_{11} & a_{12}\\a_{21} & a_{22}\end{bmatrix} }\)

Matrix \({\ B = \begin{bmatrix}b_{11} & b_{12}\\b_{21} & b_{22}\end{bmatrix} }\)

The result of adding A to B is a new matrix C.

Matrix \({\ C = \begin{bmatrix}c_{11} & c_{12}\\c_{21} & c_{22}\end{bmatrix} = A+B = \begin{bmatrix}a_{11} & a_{12}\\a_{21} & a_{22}\end{bmatrix} + \begin{bmatrix}b_{11} & b_{12}\\b_{21} & b_{22}\end{bmatrix} }\)

where each element \({\ c_{ij} = a_{ij} + b_{ij} }\).

\({\ C = \begin{bmatrix}c_{11} & c_{12}\\c_{21} & c_{22}\end{bmatrix} = \begin{bmatrix}a_{11} + b_{11} & a_{12} + b_{12}\\a_{21} + b_{21} & a_{22} + b_{22}\end{bmatrix} }\)

The explanation holds for any other matrix dimension.

Addition of 3x3 Matrices

Consider two matrices A and B of dimension 3x3:

Matrix \({\ A = \begin{bmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} &a_{23}\\a_{31} & a_{32} &a_{33}\end{bmatrix} }\)

Matrix \({\ B = \begin{bmatrix}b_{11} & b_{12} & b_{13}\\b_{21} & b_{22} &b_{23}\\b_{31} & b_{32} &b_{33}\end{bmatrix} }\)

The result of adding A to B is a new matrix C.

Matrix \({\ C = \begin{bmatrix}c_{11} & c_{12} & c_{13}\\c_{21} & c_{22} &c_{23}\\c_{31} & c_{32} &c_{33}\end{bmatrix} = \begin{bmatrix}a_{11} + b_{11} & a_{12} + b_{12} & a_{13} + b_{13}\\a_{21} + b_{21} & a_{22} + b_{22} & a_{23} + b_{23}\\a_{31} + b_{31} & a_{32} + b_{32} & a_{33} + b_{33}\end{bmatrix} }\)

Solved Examples - Matrix Addition

1. Find the sum of following two matrices: A and B.

Matrix \({\ A = \begin{bmatrix}7 & 2\\3 & 1\end{bmatrix} }\)

Matrix \({\ B = \begin{bmatrix}4 & 5\\4 & 0\end{bmatrix} }\)

Solution

2. Find the sum of following two matrices: A and B.

Matrix \({\ A = \begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\end{bmatrix} }\)

Matrix \({\ B = \begin{bmatrix}10 & 10 & 10\\10 & 10 & 10\end{bmatrix} }\)

Solution

3. Find the sum of following two matrices: A and B.

Matrix \({\ A = \begin{bmatrix}1 & 2\\4 & 5\end{bmatrix} }\)

Matrix \({\ B = \begin{bmatrix}10 & 10 & 10\\10 & 10 & 10\end{bmatrix} }\)

Solution

Properties of Matrix Addition

Addition operation on Matrices obey these properties.

1. Commutative Property

Commutative Property of Addition operation states that the result of addition of matrices does not depend on the position of matrices in the addition expression.

If A and B are two given matrices of same size, then according to the commutative property of addition,

\({\ A + B = B + A }\)

2. Associative Property

Associative Property of Addition operation states that the result of addition does not depend on the order of addition.

If A, B, and C are three given matrices of same size, then according to the associative property of addition,

\({\ A + (B + C) = (A + B) + C }\)

On the left hand side, B and C are added first, and then their result is added to A.

On the right hand side, A and B are added first, and then their result is added to C.

3. Additive Identity

Additive Identity states that for any matrix A, there exists a matrix B such that

\({\ A + B = A }\)

For the above statement to be true, matrix B should be a zero matrix of the size same as that of matrix A.

\({\ A + O = A }\)

where \({\ O }\) is zero matrix.

Therefore, Zero matrix is the Additive Identity of matrix A.

4. Additive Inverse

Additive Inverse property states that for any matrix A, there exists a matrix B such that

\({\ A + B = O }\)

For the above statement to be true, matrix B should be negative of matrix A.

\({\ B = -A }\)

Therefore, matrix: -A is the Additive Inverse of matrix A.