Problem 85 - 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

The matrices satisfy the associative property: \((A B) C = A (B C)\).

Step 1: Define Each Matrix and Expression

We are given three matrices: \( A = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \), \( B = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} \), \( C = \begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{bmatrix} \).\ We want to show that \((A B) C = A(B C)\).

Step 2: Compute \(A \, B\)

First, we calculate the product \( A B \). Multiplying the two matrices, we have:\[A B = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \cdot \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix}\]

Step 3: Compute \((A B) \, C\)

Using our result from Step 2, we multiply \( (A B) \) with \( C \):\[(A B) C = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix} \cdot \begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{bmatrix}\]This results in:\[\begin{bmatrix} (a_{11}b_{11} + a_{12}b_{21})c_{11} + (a_{11}b_{12} + a_{12}b_{22})c_{21} & (a_{11}b_{11} + a_{12}b_{21})c_{12} + (a_{11}b_{12} + a_{12}b_{22})c_{22} \ (a_{21}b_{11} + a_{22}b_{21})c_{11} + (a_{21}b_{12} + a_{22}b_{22})c_{21} & (a_{21}b_{11} + a_{22}b_{21})c_{12} + (a_{21}b_{12} + a_{22}b_{22})c_{22} \end{bmatrix}\]

Step 4: Compute \(B \, C\)

Now, let's calculate \( B C \):\[B C = \begin{bmatrix} b_{11} & b_{12} \ b_{21} & b_{22} \end{bmatrix} \cdot \begin{bmatrix} c_{11} & c_{12} \ c_{21} & c_{22} \end{bmatrix} = \begin{bmatrix} b_{11}c_{11} + b_{12}c_{21} & b_{11}c_{12} + b_{12}c_{22} \ b_{21}c_{11} + b_{22}c_{21} & b_{21}c_{12} + b_{22}c_{22} \end{bmatrix}\]

Step 5: Compute \(A \, (B \, C)\)

Using the result from Step 4, we now multiply \( A \) with \( (B C) \):\[A (B C) = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \cdot \begin{bmatrix} b_{11}c_{11} + b_{12}c_{21} & b_{11}c_{12} + b_{12}c_{22} \ b_{21}c_{11} + b_{22}c_{21} & b_{21}c_{12} + b_{22}c_{22} \end{bmatrix}\]This results in:\[\begin{bmatrix} a_{11}(b_{11}c_{11} + b_{12}c_{21}) + a_{12}(b_{21}c_{11} + b_{22}c_{21}) & a_{11}(b_{11}c_{12} + b_{12}c_{22}) + a_{12}(b_{21}c_{12} + b_{22}c_{22}) \ a_{21}(b_{11}c_{11} + b_{12}c_{21}) + a_{22}(b_{21}c_{11} + b_{22}c_{21}) & a_{21}(b_{11}c_{12} + b_{12}c_{22}) + a_{22}(b_{21}c_{12} + b_{22}c_{22}) \end{bmatrix}\]

Step 6: Compare Results from Step 3 and Step 5

By simplifying and comparing the final matrices obtained from \((A B) C\) and \(A (B C)\), you'll find that they match element-wise due to equivalency transformations involving associative properties of real number operations. Hence, we conclude \((A B) C = A (B C)\).

Key Concepts

Associative Property in Matrix Algebra

Understanding the associative property is crucial in matrix algebra, especially when dealing with operations on matrices. In general mathematics, the associative property allows you to regroup terms in a product without affecting the result.
In the context of matrix multiplication, this property states that when we have three matrices, say \(A\), \(B\), and \(C\), the product \((A \times B) \times C\) is equal to the product \(A \times (B \times C)\).

• This means you can multiply \(A\) and \(B\) first, and then multiply that product with \(C\). Or, you can multiply \(B\) and \(C\) first, and then multiply \(A\) with that result.
• The associative property makes calculations more flexible and can simplify complex problems by allowing us to choose the order of multiplication that is most convenient.
• However, note that while the associative property applies to matrix multiplication, it does not apply to addition or subtraction of matrices in the same way.

Recognizing and applying this property is essential for manipulating larger matrices efficiently.

Matrix Multiplication Basics

Matrix multiplication might seem complicated at first, but understanding the basics can make it easier. It's a process that involves calculating the dot product of rows from the first matrix with columns of the second matrix.
Here's how you can multiply two matrices:

• Take each element of a row from the first matrix \(A\) and multiply it by its corresponding element in a column of the second matrix \(B\).
• Add results together to get the new element in the resulting matrix at a specific position.
• Repeat this process for all rows and columns.

When multiplying \(2\times2\) matrices, each item in the resulting matrix needs to be calculated separately. This requires performing summations of these products for every entry according to position. Thus, matrices \(A\), \(B\), and \(C\) can be multiplied to prove the associative property by calculating \((A \times B) \times C\) and \(A \times (B \times C)\) as illustrated in the exercise.
This thorough approach not only shows the associative property holds but also reinforces the mechanical process of matrix multiplication.

Understanding 2x2 Matrices

Let's focus on \(2 \times 2\) matrices, a simple yet powerful foundation in matrix algebra. A \(2\times2\) matrix has two rows and two columns, forming a square matrix structure. Each entry is usually denoted by combinations of row and column indices, like \( a_{11}, a_{12}, a_{21}, a_{22}\) for matrix \(A\).
Why are \(2\times2\) matrices important?

• They are the simplest form of square matrices, often used for introducing concepts such as determinants, inverses, and eigenvalues.
• Understanding \(2 \times 2\) matrices makes it easier to extend learning to larger matrices.
• They are commonly used in systems of linear equations, transformations, and various applications like computer graphics.

In the exercise, we dealt with matrices \(A\), \(B\), and \(C\) each being \(2\times2\). This simplicity allowed us to manually verify the associative property through step-by-step matrix multiplication without the overwhelming complexity that larger matrices might introduce.