Problem 83 - A Graphical Approach to College Algebra

Question

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. 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

\(A+B=B+A\) holds due to the commutative property of real number addition.

Step 1: Define matrix addition

Matrix addition involves adding corresponding elements from each matrix. For matrices \( A \) and \( B \), the result is a matrix \( A+B \) whose \( (i,j) \)-th element is \( a_{ij} + b_{ij} \). The addition of \( A \) and \( B \) is given by: \[A+B = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}\]

Step 2: Define matrix addition in reverse order

Similarly, the matrix \( B+A \) is the result of adding matrices \( B \) and \( A \) together, resulting in a matrix where the \( (i,j) \)-th element is \( b_{ij} + a_{ij} \). This leads to:\[B+A = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} + \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} = \begin{bmatrix} b_{11} + a_{11} & b_{12} + a_{12} \ b_{21} + a_{21} & b_{22} + a_{22} \end{bmatrix}\]

Step 3: Apply the commutative property of real numbers

Since the elements of matrices \( A \) and \( B \) are real numbers, the commutative property of addition for real numbers states that \( a_{ij} + b_{ij} = b_{ij} + a_{ij} \) for each element. Therefore, each corresponding element of \( A+B \) and \( B+A \) will be 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 4: Conclude the proof of the commutative property for matrices

Since each element of the matrix \( A+B \) is equal to the corresponding element of matrix \( B+A \), the entire matrices are equal: \( A+B = B+A \). This confirms that the commutative property holds for the addition of \( 2 \times 2 \) matrices.

Key Concepts

Commutative Property

The commutative property is a fundamental concept in mathematics, especially in matrix operations. It states that the order in which you add numbers doesn't change their sum. For real numbers, this means that if you have two numbers, say 5 and 8, then 5 + 8 is the same as 8 + 5. This same idea extends to matrices when you are adding them.

In the context of matrices, particularly 2x2 matrices, this means if you have two matrices A and B, then adding them one way (A + B) gives you the same result as adding them the other way (B + A). The commutative property ensures that the result is unaffected by the order of the matrices in this operation.

2x2 Matrices

A 2x2 matrix is a specific type of matrix that contains two rows and two columns. Each position in the matrix is filled with an element, often denoted by variables or numbers. The general form of a 2x2 matrix looks like this: \[\begin{bmatrix}a_{11} & a_{12} \a_{21} & a_{22}\end{bmatrix}\]

Here, \(a_{11}\), \(a_{12}\), \(a_{21}\), and \(a_{22}\) represent the elements of the matrix. The indices represent the row and column positions respectively. This structure is compact yet powerful, making it a useful tool in various fields including computer graphics, physics, and statistics.

• Useful for representing transformations.
• Provides a basic structure for solving systems of linear equations.
• Allows for simple calculations involving matrix operations.

Matrix Operations

Matrix operations form the backbone of many mathematical computations and involve various techniques like addition, subtraction, multiplication, and more. Each operation follows specific rules that dictate how matrices interact with one another.

Opting for addition, the process involves combining two matrices of the same dimensions by adding their respective components or entries. If you have matrix \(A\) and matrix \(B\), both 2x2 matrices, the addition is performed element-wise. That means you take the element from the first position of \(A\) and add it to the first position of \(B\), and so on. The notation of adding matrices \(A\) and \(B\) is expressed as:\

\[A + B = \begin{bmatrix} a_{11}+b_{11} & a_{12}+b_{12} \ a_{21}+b_{21} & a_{22}+b_{22} \end{bmatrix}\]

• Matrix operations follow a defined structure and rules.
• They are crucial for calculating solutions in physics and engineering.
• Provide the method to solve complex mathematical problems efficiently.

Following these operations correctly ensures accurate results in various applications where matrices are used.