Matrix Addition and Subtraction

A matrix is a rectangular array or table of numbers, symbols, or expressions organized in rows and columns that are used to represent a mathematical object.

• Square matrix, null matrix, diagonal matrix, scalar matrix, identity matrix and diagonal matrix are different types of matrix.
• Matrix addition refers to the operation of adding entries of two or more matrices together.
• The addition of matrices is possible only if the order of the matrices is the same.
• Matrices represent linear mappings of numbers and allow explicit calculations in linear algebra without any further requirements.
• Matrix addition is used in the fields of data analysis, image processing, cryptography, operations research and machine learning.

Key Terms: Matrix, Matrix Addition, Matrix Subtraction, Properties of Matrix Addition, Commutative Property, Associative Property, Additive Identity, Additive Inverse

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What is Matrix Addition?

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A matrix refers to a set of numbers arranged in rows (m) and columns (n) to form a rectangular array of m x n. The dimension or size of a matrix is the total number of rows and columns in an allocated matrix.

• Many operations, such as addition, subtraction, scalar multiplication and transposition may be performed on matrices. 
• The action of adding equivalent elements of two or more matrices is known as matrix addition. 
• Matrix addition can be done only for matrices of the same size (or dimension).
• The element-wise addition of matrices and the direct sum of matrices are two methods for adding matrices.

The matrix addition is defined by two requirements in general, which are as follows:

• Consider the following two matrices: A and B.
• These matrices can be combined if (and only if) their order is identical, that is, the two matrices have the same number of rows and columns. 
• If, for example, matrix A has an order of 3*4, then matrix B can be added to it if its order is likewise 3*4.
• For matrices of variable sizes, the addition of matrices is not specified.

Consider two matrices A =[aij]_m*n and B =[bij]_m*n of order m * n, then the addition of A and B is given by the formula;

A + B = [a_ij]_m*n + [b_ij]_m*n = [a_ij + b_ij]_m*n

How to add Matrices?

To add two or more matrices, the addition of a matrix is the most basic matrix operation. The addition of matrices is only feasible if the provided matrices are in the same order.

• Look at some specific examples to gain a better understanding.

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Properties of Matrix Addition

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The fundamental features of matrix addition are comparable to those of real-number addition. Assume A, B, and C are three m*n matrices. The matrix addition operation has the following characteristics.

Commutative Property

When A and B are two matrices of the same order, such as m * n, then the addition of the two matrices is commutative only if A + B = B + A.

Associative Property

If A, B, and C are three matrices of the same order, such as m * n, then their addition is associative, i.e., A + (B + C) = (A + B) + C.

Additive Identity

There is an identity element in any m x n matrix. As a result, if A is a m x n order matrix, A's additive identity will be zero matrices of the same order, resulting in A + O = A ( where O is an additive identity)

Additive Inverse

If A is an m x n matrix, then B (= -A) of the same order is the additive inverse of A, resulting in A + B = O. Therefore, the total of the matrices and their additive inverses yields a zero matrix.

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What is Matrix Subtraction?

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Matrix subtraction or subtraction of matrix is defined as the process of addition of the negative of a matrix to another matrix. Like matrix addition, matrix subtraction is possible if order of both the matrix are similar.

• Suppose there are two matrices, A = [a_ij] and B = [b_ij] of the same order, then the subtraction of A and B, i.e., A – B is defined as: A – B = a_ij – b_ij

• The matrix subtraction does not follow commutative law which means A – B ≠ B – A.

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Solved Examples on Matrix Addition

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The examples on Matrix addition are as follows:

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Things to Remember

• A matrix is a two-dimensional rectangular array of integers or symbols organized in rows and columns.
• The number of rows and columns determines the matrix's order. 
• The numbers in the matrix are the entries; each number is referred to as an element. Matrix is the plural form of a matrix.
• Matrix addition is a form of operation performed to add two matrices to get a single matrix.
• In electrical circuits, matrices are often employed to solve AC network equations.
• The elements of the matrices are added when matrices are of the same order.