Problem 44 - Algebra for College Students

Question

How would you show that addition of \(2 \times 2\) matrices is a commutative operation?

Step-by-Step Solution

Answer

Addition of \(2 \times 2\) matrices is commutative.

Step 1: Understanding the Problem

The task requires us to prove that the addition of two \(2 \times 2\) matrices is commutative. This means we need to show that for any two \(2 \times 2\) matrices \(A\) and \(B\), \(A + B = B + A\).

Step 2: Defining the Matrices

Let \(A\) and \(B\) be two arbitrary \(2 \times 2\) matrices. Define them as follows: \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) and \(B = \begin{bmatrix} e & f \ g & h \end{bmatrix}\).

Step 3: Adding the Matrices

Perform the addition of matrices \(A\) and \(B\). \( A + B = \begin{bmatrix} a & b \ c & d \end{bmatrix} + \begin{bmatrix} e & f \ g & h \end{bmatrix} = \begin{bmatrix} a+e & b+f \ c+g & d+h \end{bmatrix}.\)

Step 4: Switching the Matrices

Now, perform the addition of matrices \(B\) and \(A\) in reverse order. \( B + A = \begin{bmatrix} e & f \ g & h \end{bmatrix} + \begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} e+a & f+b \ g+c & h+d \end{bmatrix}.\)

Step 5: Showing Equality

Compare the two results: The element-wise sums \(a+e\) and \(e+a\), \(b+f\) and \(f+b\), \(c+g\) and \(g+c\), \(d+h\) and \(h+d\) are equal by the commutative property of addition for real numbers. Thus, \( \begin{bmatrix} a+e & b+f \ c+g & d+h \end{bmatrix} = \begin{bmatrix} e+a & f+b \ g+c & h+d \end{bmatrix}. \)

Step 6: Conclusion

Since \(A + B = B + A\), this demonstrates that the addition of \(2 \times 2\) matrices is indeed commutative.

Key Concepts

Commutative Property

The commutative property is a fundamental aspect of mathematics, particularly when it comes to operations like addition and multiplication. When we say an operation is commutative, we mean that changing the order of the operands does not change the result.
For example, with the commutative property of addition, we have:

• For numbers: If you take two numbers, such as 3 and 5, and add them together, the order doesn't matter: \(3 + 5 = 5 + 3\).
• For matrices: We use the same concept for matrix addition. If you have two matrices \(A\) and \(B\), adding \(A + B\) will yield the same result as \(B + A\).

This property simplifies many mathematical procedures and calculations because you can freely rearrange terms without affecting outcomes.

2x2 Matrices

Matrices are rectangular arrays of numbers that are invaluable in various areas of mathematics and science. A \(2 \times 2\) matrix specifically consists of four elements, organized in two rows and two columns.
Here's an example of a \(2 \times 2\) matrix:

• \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\)

In the matrix above, \(a, b, c,\) and \(d\) are elements which can be managed using various matrix operations, such as addition, subtraction, and multiplication. Understanding the organization and handling of such matrices is important for applying them in fields like algebra, statistics, physics, and computer science.

Matrix Operations

Matrix operations serve as the foundation for many advanced calculations and applications in both theoretical and applied mathematics. Two basic operations with matrices are addition and subtraction.
When adding matrices, each element of one matrix is added to its corresponding element of the other matrix. Let's consider matrices \(A\) and \(B\) as follows:

• \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\)
• \(B = \begin{bmatrix} e & f \ g & h \end{bmatrix}\)

The result of \(A + B\) will be:

• \(A + B = \begin{bmatrix} a+e & b+f \ c+g & d+h \end{bmatrix}\)

In this operation, you're simply performing addition on a component-by-component basis. It is crucial to note that operations such as these require matrices of the same dimension, ensuring each element has a corresponding pair.

Algebraic Proof

An algebraic proof is a logical series of statements used to demonstrate the truth of a given hypothesis using algebraic principles. In the context of matrix addition, proving that matrix addition is commutative involves a series of algebraic steps to show that \(A + B = B + A\).
The proof starts by defining arbitrary matrices \(A\) and \(B\):

• \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\)
• \(B = \begin{bmatrix} e & f \ g & h \end{bmatrix}\)

Then we show:

• \(A + B = \begin{bmatrix} a+e & b+f \ c+g & d+h \end{bmatrix}\)
• \(B + A = \begin{bmatrix} e+a & f+b \ g+c & h+d \end{bmatrix}\)

Based on the commutative property of real numbers, the equalities \(a+e = e+a\), \(b+f = f+b\), \(c+g = g+c\), and \(d+h = h+d\) hold true.
Thus, it proves that \(A + B = B + A\), confirming the commutative nature of matrix addition.