Problem 27 - Discrete Mathematics with Applications

Question

Let \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\) be any \(m \times n\) matrices, \(O\) the \(m \times n\) zero matrix, and \(c\) and \(d\) any real numbers. Prove each (see Theorem 3.12 ). $$A+B=B+A$$

Step-by-Step Solution

Answer

Given two m x n matrices A and B, with elements denoted as \(a_{ij}\) and \(b_{ij}\), we want to prove that \(A + B = B + A\). We perform the matrix addition operation element-wise: \(A + B = (a_{ij} + b_{ij})\) and \(B + A = (b_{ij} + a_{ij})\) Since the commutative property of real numbers states that \(a_{ij} + b_{ij} = b_{ij} + a_{ij}\), we can conclude that matrix addition is commutative: \(A + B = B + A\)

Step 1: Define the matrices A and B

Given two m x n matrices A and B, where each element of matrix A is denoted as \(a_{ij}\) and each element of matrix B is denoted as \(b_{ij}\), we want to prove that \(A + B = B + A\).

Step 2: Perform the addition operation element-wise

We perform the matrix addition operation, adding the corresponding elements of matrices A and B: \(A + B = (a_{ij} + b_{ij})\), where i = 1, 2, ..., m and j = 1, 2, ..., n Similarly, we perform the matrix addition operation in reverse order: \(B + A = (b_{ij} + a_{ij})\), where i = 1, 2, ..., m and j = 1, 2, ..., n

Step 3: Show that the sums are equal

Now, we need to show that \(a_{ij} + b_{ij} = b_{ij} + a_{ij}\). This is true because, based on commutative property of real numbers, we know that two real numbers a and b satisfy \(a + b = b + a\). Therefore, since \(a_{ij}\) and \(b_{ij}\) are real numbers: \(a_{ij} + b_{ij} = b_{ij} + a_{ij}\)

Step 4: Conclusion

Based on the commutative property of real numbers, we have shown that each element of the sum of the two matrices A and B is equal to the corresponding element of the sum of the matrices B and A. Therefore, we can conclude that matrix addition of A and B is commutative: \(A + B = B + A\)

Key Concepts

Discrete Mathematics

Discrete mathematics is a branch of mathematics primarily concerned with countable, distinct elements and the relationships between them. Unlike its counterpart, continuous mathematics, which deals with continuous variables, discrete mathematics focuses on structures that are inherently separate and finite.
Within discrete mathematics, we explore concepts such as graph theory, logic, set theory, and combinatorics, among others. Matrix operations, which are part of linear algebra, also fall under the umbrella of discrete mathematics, because they involve manipulating arrays of numbers that are discrete and finite in nature. Understanding matrix operations is crucial for solving a variety of computing problems, especially in fields like computer science, cryptography, and network theory.

Commutative Property

The commutative property is a foundational concept in mathematics and asserts that the order in which two elements are combined under a given binary operation does not affect the end result. In the context of addition, this means that if we have two elements, say a and b, then a + b will always equal b + a. This principle is not exclusive to numbers but applies to various mathematical structures, including matrices.

When dealing with matrix addition, the commutative property ensures that the matrices can be added in any order, and the resultant matrix will remain the same. This property is inherently understood in the context of real numbers, but its application to matrices is an important concept that students in discrete mathematics need to understand.

Matrix Operations

Matrix operations encompass a wide variety of computations with matrices, including addition, subtraction, multiplication, and transformation. These operations are integral to linear algebra and are utilized to solve systems of equations, transform geometric figures, and model data.

For most students, the first encounter with matrix operations involves learning how to add, subtract, and multiply matrices. These operations often involve an element-wise approach where calculations are performed on each corresponding element of the matrices. When we perform matrix addition, as in the problem at hand, each element of one matrix is added to the corresponding element of another matrix provided that both matrices have the same dimension.

Element-wise Addition

Element-wise addition is a specific type of matrix operation where two matrices of the same size are added together by combining corresponding elements. This process requires that both matrices have the same dimensions; that is, they must have the same number of rows (m) and columns (n).
For instance, if we have two matrices A and B, we find the sum matrix S by adding each element a_ij of A to the corresponding element b_ij of B to get the element s_ij of the sum matrix S (i.e., s_ij = a_ij + b_ij). As part of the commutative property discussed earlier, the order in which we perform this addition does not affect the result, meaning that A + B is the same as B + A at every corresponding element.