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

Step-by-Step Solution

Answer

Matrix addition is associative; both expressions are equal.

Step 1: Matrix Addition of B and C

First, add matrices \(B\) and \(C\). To do this, add the corresponding elements of the matrices: \[B + C = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} + \begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{bmatrix} = \begin{bmatrix} b_{11} + c_{11} & b_{12} + c_{12} \ b_{21} + c_{21} & b_{22} + c_{22} \end{bmatrix}\]

Step 2: Add A to (B+C)

Now, we add matrix \(A\) to the expression \((B + C)\). We do this by adding each corresponding element: \[A + (B + C) = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} + \begin{bmatrix} b_{11} + c_{11} & b_{12} + c_{12} \ b_{21} + c_{21} & b_{22} + c_{22} \end{bmatrix} = \begin{bmatrix} a_{11} + (b_{11} + c_{11}) & a_{12} + (b_{12} + c_{12}) \ a_{21} + (b_{21} + c_{21}) & a_{22} + (b_{22} + c_{22}) \end{bmatrix}\]

Step 3: Matrix Addition of A and B

Now, add matrices \(A\) and \(B\) to start forming \((A + B) + C\). Add the corresponding elements: \[A + B = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} + \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}\]

Step 4: Add C to (A+B)

Finally, add matrix \(C\) to the result obtained in the last step: \[(A + B) + C = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix} + \begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{bmatrix} = \begin{bmatrix} (a_{11} + b_{11}) + c_{11} & (a_{12} + b_{12}) + c_{12} \ (a_{21} + b_{21}) + c_{21} & (a_{22} + b_{22}) + c_{22} \end{bmatrix}\]

Step 5: Final Verification

Compare the results from previous steps: \[A + (B + C) = \begin{bmatrix} a_{11} + (b_{11} + c_{11}) & a_{12} + (b_{12} + c_{12}) \ a_{21} + (b_{21} + c_{21}) & a_{22} + (b_{22} + c_{22}) \end{bmatrix}\] and \[(A + B) + C = \begin{bmatrix} (a_{11} + b_{11}) + c_{11} & (a_{12} + b_{12}) + c_{12} \ (a_{21} + b_{21}) + c_{21} & (a_{22} + b_{22}) + c_{22} \end{bmatrix}\]. Since matrix addition is associative, these two are equal, verifying the associative property \(A + (B + C) = (A + B) + C\).

Key Concepts

Associative Property

The Associative Property is a fundamental principle of mathematics that simplifies calculations by altering the grouping of operations without affecting the outcome. In the context of matrix addition, it states that the way in which matrices are grouped during addition doesn't change the final result. This property can be written as \( A + (B + C) = (A + B) + C \). Here, the matrices \( A \), \( B \), and \( C \) can be added in different orders without impacting the sum.

This property is important in ensuring that mathematical expressions remain consistent regardless of grouping. With matrices, when you apply this property, you verify that the sum remains the same, making calculations more flexible and easier to manage, especially when handling complex computations.

• It allows for parentheses to change position during calculations.
• Enables simplification of matrix expressions.
• Ensures stability and uniformity in matrix arithmetic operations.

Understanding and applying the associative property in matrices gives students a significant advantage in solving matrix-related problems efficiently.

2x2 Matrices

A 2x2 matrix is a rectangular array with two rows and two columns. This type of matrix is one of the simplest forms in the matrix family. Each element of a 2x2 matrix is often denoted by a unique subscript to identify its position within the matrix. For instance, a generic 2x2 matrix \( A \) can be represented as:

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

These matrices are important in many areas including computer graphics, systems of equations, and transformations in geometry. Their simplicity makes them ideal for introductory lessons in matrix algebra. When performing operations like matrix addition, corresponding elements from each matrix are added together. This straightforward process is foundational for understanding more complex matrix operations.

Since 2x2 matrices are used frequently:

• They're a good starting point for learning about matrices and their properties.
• They aid in resolving systems with two linear equations.
• They help in understanding the basic structure and manipulation of matrix data.

Recognizing these matrices and understanding their manipulation prepares students for more advanced topics like determinants and matrix multiplication.

Matrix Properties

Matrix Properties encompass several rules that govern operations on matrices, simplifying complex problems and ensuring consistency. In addition to the Associative Property, several other key properties are crucial in matrix arithmetic.

The **Commutative Property** states that the order of addition doesn't affect the sum; this can be written as \( A + B = B + A \). However, this property does not apply to matrix multiplication.

**Identity Matrices** act as the additive identity in matrix addition. Adding an identity matrix to any matrix leaves the original matrix unchanged. Denoted as matrix \( I \), it has ones on its diagonal and zeros elsewhere:

\[I = \begin{bmatrix}1 & 0 \0 & 1\end{bmatrix}\]These properties help in simplifying expressions and solving equations within linear algebra.

In matrix operations strategy:

• Associative and commutative properties are crucial for understanding and simplification.
• Identity matrices are vital in transformations and as reference points.
• Understanding these properties equips students to handle complex matrix equations confidently.

Mastering these matrix properties ensures robust mathematical reasoning and problem-solving capabilities.