Problem 83 - A Graphical Approach to Precalculus with Limits

Question

Let $$A=\left[\begin{array}{ll}a_{11} & a_{12} \\\a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll} b_{11} & b_{12} \\\b_{21} & b_{22}\end{array}\right], \text { and } C=\left[\begin{array}{ll}c_{11} & c_{12} \\\c_{21} & c_{22}\end{array}\right]$$ where all the elements are real mumbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(A+B=B+A\) (commutative property)

Step-by-Step Solution

Answer

Matrix addition is commutative; hence, \(A + B = B + A\).

Step 1: Understanding the Commutative Property

The commutative property states that the order in which two numbers (or matrices) are added does not affect the sum. Mathematically, this is expressed as \(A + B = B + A\) for matrices.

Step 2: Setting Up Matrix Addition

To apply the commutative property to the given matrices, \(A\) and \(B\), we need to perform matrix addition. Recall that for two matrices to be addable, they must be of the same dimension. Here, both \(A\) and \(B\) are \(2 \times 2\) matrices.

Step 3: Calculate \(A + B\)

To find \(A + B\), add the corresponding elements of matrices \(A\) and \(B\):\[A + B = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}\]

Step 4: Calculate \(B + A\)

Now calculate matrix \(B + A\) by adding the corresponding elements of \(B\) and \(A\) in reverse order:\[B + A = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} + \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} b_{11} + a_{11} & b_{12} + a_{12} \ b_{21} + a_{21} & b_{22} + a_{22} \end{bmatrix}\]

Step 5: Simplifying and Comparing Results

Notice that addition of real numbers is commutative, meaning \(a_{ij} + b_{ij} = b_{ij} + a_{ij}\) for each corresponding element. Therefore, the resulting matrices from both operations, \(A + B\) and \(B + A\), are equal:\[\begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix} = \begin{bmatrix} b_{11} + a_{11} & b_{12} + a_{12} \ b_{21} + a_{21} & b_{22} + a_{22} \end{bmatrix}\]

Step 6: Conclusion of the Commutative Property

Since the matrices \(A + B\) and \(B + A\) are equal, we have shown that matrix addition is commutative for \(2 \times 2\) matrices, as expected from the comitative property of addition.

Key Concepts

Commutative Property

The commutative property is a fundamental principle in mathematics stating that the order in which two numbers or expressions are added does not affect the outcome. This property applies to both numbers and matrices. Hence, for any two matrices, let's say \( A \) and \( B \), the relation \( A + B = B + A \) holds true.
When performing matrix addition, this property ensures that changing the order of matrices does not change the resulting matrix. This concept mirrors the commutative property of real numbers where, for example, \( 3 + 4 \) is the same as \( 4 + 3 \). Similarly, in matrix operations, each corresponding element in the matrix \( A \) adds to its counterpart in matrix \( B \) in any order. As a result:

• The position doesn't matter when adding matrices.
• The end result remains unchanged, embodying the commutative nature.

Understanding this concept is crucial as it forms the basis of more complex matrix operations.

2x2 Matrices

A 2x2 matrix is a simple type of matrix with two rows and two columns. It is an essential part of linear algebra and is often one of the first matrices introduced.
A matrix like this could be represented as:
\[ A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \]
Here:

• Each entry such as \( a_{11} \) or \( a_{21} \) is a real number.
• The numbers refer to the position in the matrix (row and column).
• The first number in the subscript denotes the row, and the second the column.

Matrix operations often involve these positions quite a bit, especially in addition, multiplication, and inversion. The uniformity of 2x2 matrices makes them ideal for illustrating properties, like the commutative property, due to their simple layout and size.

Real Numbers

Real numbers include all the numbers on the continuous number line, encompassing integers, fractions, and irrational numbers. In matrices, elements such as \( a_{11} \) and \( b_{22} \) are real numbers. This is significant in matrix operations, ensuring each entry in the matrix is defined and the result remains within the field of real numbers.
Real numbers ensure:

• Predictable outcomes for addition and multiplication.
• Each operation within a matrix remains consistent, stable, and applicable across all dimensions.
• Elements can be compared easily, aiding calculations in both theoretical and practical applications.

This consistency of real numbers is vital when verifying properties such as the commutative property in matrix addition, ensuring that operations on matrix elements do not deviate into undefined behaviors.

Matrix Operations

Matrix operations include a variety of processes like addition, subtraction, multiplication, and more. Each operation follows specific rules that ensure the results are valid within the context of linear algebra.
When adding matrices, like those provided in the example, you:

• Add corresponding elements together. For a 2x2 matrix use \( a_{ij} + b_{ij} \).
• Ensure matrices are of the same size to perform valid addition.
• Apply properties such as commutativity to check and simplify results.

Each of these processes makes use of the properties inherent in real numbers and matrix dimensions. This assures proper and logical outcomes—an understanding of these core concepts is pivotal for performing and verifying matrix operations correctly. This systematic approach helps solve complex problems by breaking them into smaller, manageable parts.