Problem 36 - Algebra for College Students

Question

For Problems \(35-43\), use the following matrices. \( \begin{aligned} A &=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] & B &=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right] \\ C &=\left[\begin{array}{ll} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\right] & O &=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right] \end{aligned} \) Show that \((A+B)+C=A+(B+C)\).

Step-by-Step Solution

Answer

Matrix addition is associative, so ((A+B)+C)=A+(B+C).

Step 1: Understand Matrix Addition

In matrix addition, matrices of the same dimensions are added by simply adding their corresponding elements. For example, if \( A = \left[ \begin{array}{cc} a_{11} & a_{12} \ a_{21} & a_{22} \end{array} \right] \) and \( B = \left[ \begin{array}{cc} b_{11} & b_{12} \ b_{21} & b_{22} \end{array} \right] \), then \( A + B = \left[ \begin{array}{cc} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{array} \right] \).

Step 2: Add A and B Together

Calculate \( A + B \) by adding their corresponding elements. Thus, \( A + B = \left[ \begin{array}{cc} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{array} \right] \).

Step 3: Add Result to C

Now add matrix \( C \) to \( A + B \). The matrix \( C \) is \( \left[ \begin{array}{cc} c_{11} & c_{12} \ c_{21} & c_{22} \end{array} \right] \). This yields \( (A+B) + C = \left[ \begin{array}{cc} (a_{11} + b_{11}) + c_{11} & (a_{12} + b_{12}) + c_{12} \ (a_{21} + b_{21}) + c_{21} & (a_{22} + b_{22}) + c_{22} \end{array} \right] \).

Step 4: Add B and C Together

Calculate \( B + C \) by adding the corresponding elements. This gives \( B + C = \left[ \begin{array}{cc} b_{11} + c_{11} & b_{12} + c_{12} \ b_{21} + c_{21} & b_{22} + c_{22} \end{array} \right] \).

Step 5: Add A to Result

Finally, add \( A \) to \( B + C \). This results in \( A + (B + C) = \left[ \begin{array}{cc} a_{11} + (b_{11} + c_{11}) & a_{12} + (b_{12} + c_{12}) \ a_{21} + (b_{21} + c_{21}) & a_{22} + (b_{22} + c_{22}) \end{array} \right] \).

Step 6: Compare the Results

Compare the matrices from Steps 3 and 5. Notice that both \((A+B)+C\) and \(A+(B+C)\) yield the same matrix: \( \left[ \begin{array}{cc} a_{11} + b_{11} + c_{11} & a_{12} + b_{12} + c_{12} \ a_{21} + b_{21} + c_{21} & a_{22} + b_{22} + c_{22} \end{array} \right] \). This confirms the associative property for matrix addition.

Key Concepts

Associative Property in Matrix Addition

In mathematics, the associative property plays a fundamental role in operations such as addition and multiplication. The associative property states that the way in which numbers (or elements, in the case of matrices) are grouped does not affect the sum or product. For matrices, this means that regardless of how matrices are grouped in an addition operation, the result will remain the same. This property can be expressed as:

• \((A+B)+C=A+(B+C)\)

The associative property ensures flexibility in matrix operations, especially in complex calculations where regrouping can simplify the process. To demonstrate this with matrices, consider the given matrices:

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

Adding these matrices in different groupings results in the same final matrix. This clearly shows the associative nature of matrix addition, thus confirming its reliability when performing matrix operations.

Understanding Matrix Operations

Matrix operations are calculations involving matrices, typically involving addition, subtraction, multiplication, and sometimes the computation of inverses or determinants in more advanced contexts. Adding matrices is quite simple: you just add corresponding elements of the matrices involved. To add two matrices, they must be of the same dimension. For instance, to add two 2x2 matrices like \(A\) and \(B\), you perform the following operations:

• Add the elements in the first row and first column: \(a_{11} + b_{11}\).
• Add the elements in the first row and second column: \(a_{12} + b_{12}\).
• Add the elements in the second row and first column: \(a_{21} + b_{21}\).
• Add the elements in the second row and second column: \(a_{22} + b_{22}\).

This gives the resulting matrix: \(\begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix} \).
Operations such as these are foundational for linear algebra and are used in various applications including computer graphics, engineering simulations, and data analysis.

Introduction to Matrices

Matrices are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns. They are a powerful tool in mathematics and are used to solve systems of linear equations, perform transformations, and model data in numerous fields. A matrix is defined by its dimensions: the number of rows and columns it contains. For example, a matrix with 2 rows and 2 columns is called a 2x2 matrix. The elements within a matrix are usually represented by discrete indices like \(a_{11}\), \(a_{12}\), confirming their specific positions within the matrix structure.

• A matrix can transform vectors and space, useful in computer science and engineering contexts.
• Matrices can represent complex systems and interactions in physics and business intelligence.

Understanding matrices is essential for those working in scientific fields, as well as those in finance and social sciences where data manipulation and transformation are critical. Matrices simplify complex problem-solving, enhancing computational efficiency.