Problem 84 - 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. \(A+(B+C)=(A+B)+C\) (associative property)

Step-by-Step Solution

Answer

The associative property for matrix addition is verified by showing \(A+(B+C)=(A+B)+C\) with expanded matrix elements.

Step 1: Compute B + C

\(B + C = \begin{bmatrix} b_{11}+c_{11} & b_{12}+c_{12} \\ b_{21}+c_{21} & b_{22}+c_{22} \end{bmatrix}\)

Step 2: Compute A + (B + C)

\(A + (B+C) = \begin{bmatrix} a_{11}+(b_{11}+c_{11}) & a_{12}+(b_{12}+c_{12}) \\ a_{21}+(b_{21}+c_{21}) & a_{22}+(b_{22}+c_{22}) \end{bmatrix}\)

Step 3: Compute A + B

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

Step 4: Compute (A + B) + C

\((A+B) + C = \begin{bmatrix} (a_{11}+b_{11})+c_{11} & (a_{12}+b_{12})+c_{12} \\ (a_{21}+b_{21})+c_{21} & (a_{22}+b_{22})+c_{22} \end{bmatrix}\)

Step 5: Conclude equality

Since real number addition is associative, i.e., \(a + (b + c) = (a + b) + c\) for all real numbers, each entry of \(A + (B+C)\) equals the corresponding entry of \((A+B) + C\). Therefore \(A + (B+C) = (A+B) + C\), proving the associative property of matrix addition for \(2 \times 2\) matrices.

Key Concepts

Associative Property in Matrix Addition

The associative property is a key feature in mathematics which states that the way in which numbers are grouped in an operation does not affect the result. In terms of matrix addition, this property informs us that the sum of matrices remains unchanged regardless of how the matrices are grouped within parentheses.
For matrices, the associative property of addition can be expressed as:

• \( (A + B) + C = A + (B + C) \)

This means if you add matrix \(A\) to the result of \(B + C\), or if you add \(B\) to \(C\) first and then add \(A\), the resulting matrix will be the same.
When dealing with matrices, it's crucial to ensure that all matrices involved have the same dimensions, specifically here as \(2 \times 2\) matrices. This property simplifies computations and guarantees consistent results, which is essential for solving bigger matrix-based problems more efficiently.

Understanding 2x2 Matrices

A \(2 \times 2\) matrix is a simple square matrix that is composed of two rows and two columns. They are formed using four elements, often represented as:

• \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \)

Here, each element \(a_{ij}\) refers to the entry located in the \(i\)-th row and \(j\)-th column of the matrix. This compact size makes them easy to work with as they contain fewer elements compared to larger matrices.
Working with \(2 \times 2\) matrices can provide foundational understanding before tackling larger and more complex matrices. They are extensively used in educational settings to model a variety of problems, including transformations in geometry and the solving of systems of linear equations.
The structure of \(2 \times 2\) matrices often forms the basis for understanding matrix algebra operations, making them fundamental building blocks in the field of linear algebra.

Concept of Matrix Operations

Matrix operations form the basis of linear algebra and involve a series of rules and procedures that manipulate matrices to achieve desired outcomes. In this context, basic matrix operations that are most commonly used include:

• Matrix Addition
• Matrix Subtraction
• Scalar Multiplication
• Matrix Multiplication

Matrix addition is one of the simplest operations where corresponding elements of matrices of the same size are added together to form a new matrix. In our exercise, the focus is on verifying the associative property with 2x2 matrices.
Another vital operation is matrix multiplication, which requires that the number of columns in the first matrix equals the number of rows in the second matrix. This operation is more complex than addition and follows different rules.
By understanding these operations, you can solve a wide range of mathematical problems and can analyze and interpret data, making matrix operations essential for various scientific and engineering applications.