Matrix addition and subtraction are two fundamental operations in linear algebra that can be performed on two matrices of the same dimensions. Matrix addition involves adding the corresponding elements of two matrices to obtain a new matrix of the same dimensions. That is, given two matrices A and B of the same dimensions, their sum C = A + B is a matrix of the same dimensions where each element c(i, j) in C is the sum of the corresponding elements a(i, j) and b(i, j) in A and B, respectively.
Matrix subtraction, on the other hand, involves subtracting the corresponding elements of two matrices to obtain a new matrix of the same dimensions. That is, given two matrices A and B of the same dimensions, their difference C = A – B is a matrix of the same dimensions where each element c(i, j) in C is the difference between the corresponding elements a(i, j) and b(i, j) in A and B, respectively.
In both cases, the matrices being added or subtracted must have the same dimensions. For example, given the following matrices:
In general, matrix addition and subtraction are important operations in linear algebra and have many applications in areas such as computer graphics, physics, and engineering, among others.
Conditions of matrix addition are follows
In order for two matrices to be added together, they must satisfy the following conditions:
- The two matrices must have the same dimensions. That is, they must have the same number of rows and the same number of columns.
- For each element in the two matrices, the corresponding elements must be of the same type. That is, if one element is a number, then the corresponding element in the other matrix must also be a number.
If both of these conditions are satisfied, then matrix addition is defined as follows:
Given two matrices A and B of the same dimensions, their sum, denoted as C, is a matrix of the same dimensions where each element in C is the sum of the corresponding elements in A and B. That is, if A = [a(i, j)] and B = [b(i, j)], then C = [c(i , j)], where c(i, j) = a(i, j) + b(i, j) for all i and j.
Symbolically, we can express matrix addition as C = A + B.
Properties of matrix addition
Matrix addition has several important properties. Let A, B, and C be matrices of the same dimensions. Then, the following properties hold for matrix addition:
- Commutative property: A + B = B + A. In other words, the order in which we add the matrices does not matter
- Associative property: (A + B) + C = A + (B + C). In other words, we can add the matrices in any order we like.
- Identity property: There exists a matrix 0 such that A + 0 = A for all matrices A of the same dimensions. In other words, adding the zero matrix, matrix does not change it.
- Inverse property: For any matrix A, there exists a matrix -A such that A + (-A) = 0. In other words, subtracting a matrix is equivalent to adding its negative.
- Scalar multiplication distributes over matrix addition: a(A + B) = a.A + a.B, where a is a scalar and A and B are matrices of the same dimensions. These properties make matrix addition a useful and versatile operation in linear algebra, and they are often used in various applications, such as solving systems of linear equations, linear transformations, and more.
Solved examples of matrix addition and subtraction:
Q1. Add two matrix given below:
C = A + B
Before adding we will check the order of two matrices so that we get to know whether we can add matrices A and B or not, so
Order of matrix A is 2 X 2 , Order of matrix B is 2 X 2
So order of both matrices are same now we can add them as shown below