Problem 45 - Algebra for College Students

Question

How would you show that subtraction of \(2 \times 2\) matrices is not a commutative operation?

Step-by-Step Solution

Answer

Matrix subtraction is not commutative because \( A - B \neq B - A \).

Step 1: Define Commutative Property

The commutative property states that the order in which two numbers or, in this case, matrices are subtracted results in the same value. For two matrices \(A\) and \(B\), if subtraction is commutative, then \( A - B = B - A \).

Step 2: Choose Specific Matrices

Choose two specific \(2 \times 2\) matrices. Let matrix \(A\) and \(B\) be:\[ A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}, B = \begin{pmatrix} 4 & 3 \ 2 & 1 \end{pmatrix} \].

Step 3: Subtract Matrices A and B

Subtract matrix \(B\) from matrix \(A\):\[ A - B = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} - \begin{pmatrix} 4 & 3 \ 2 & 1 \end{pmatrix} = \begin{pmatrix} 1-4 & 2-3 \ 3-2 & 4-1 \end{pmatrix} = \begin{pmatrix} -3 & -1 \ 1 & 3 \end{pmatrix} \].

Step 4: Subtract Matrices B and A

Subtract matrix \(A\) from matrix \(B\):\[ B - A = \begin{pmatrix} 4 & 3 \ 2 & 1 \end{pmatrix} - \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} = \begin{pmatrix} 4-1 & 3-2 \ 2-3 & 1-4 \end{pmatrix} = \begin{pmatrix} 3 & 1 \ -1 & -3 \end{pmatrix} \].

Step 5: Compare the Results

Compare the results of \(A - B\) and \(B - A\). Notice that:\[ A - B = \begin{pmatrix} -3 & -1 \ 1 & 3 \end{pmatrix} \] and\[ B - A = \begin{pmatrix} 3 & 1 \ -1 & -3 \end{pmatrix} \].Since \( A - B eq B - A \), matrix subtraction is not commutative.

Key Concepts

Commutative Property

The commutative property is a mathematical principle that states the order of operations does not affect the outcome. This is commonly seen in addition and multiplication, where swapping the order of operands results in the same answer. For example, with numbers, we have:

• For addition: \( a + b = b + a \)
• For multiplication: \( a \times b = b \times a \)

However, not all operations share this property. Subtraction is a classic operation where the commutative property does not hold true. In the context of matrices, we apply the same principle. If subtraction were commutative for matrices, we would expect \( A - B = B - A \).
But our exercise shows that in matrix subtraction, reversing the order of subtraction changes the result, highlighting that this operation is non-commutative. Understanding why this property fails in certain operations helps reinforce the need to carefully consider the sequence of operations in calculations.

2x2 Matrices

A 2x2 matrix is a simple yet powerful mathematical tool made up of four elements arranged in two rows and two columns: \[ A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \]Each element can be a number, and combining them in various ways gives matrices their unique properties and applications, such as transformations and system solving.
In our exercise, two specific matrices \(A\) and \(B\) are given:\[ A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 3 \ 2 & 1 \end{pmatrix} \]Subtracting these matrices involves taking the difference of corresponding elements:

• Top left: \(1 - 4 = -3\)
• Top right: \(2 - 3 = -1\)
• Bottom left: \(3 - 2 = 1\)
• Bottom right: \(4 - 1 = 3\)

This process clearly shows how each element contributes to the final result of matrix subtraction.

Non-Commutative Operation

Matrix subtraction is a non-commutative operation, which means changing the order of the matrices affects the outcome. In this operation, unlike addition or multiplication, reversing the operands produces a completely different result:

• First, consider \( A - B \): The resulting matrix is: \( \begin{pmatrix} -3 & -1 \ 1 & 3 \end{pmatrix} \)
• Then, consider \( B - A \): This gives \( \begin{pmatrix} 3 & 1 \ -1 & -3 \end{pmatrix} \)

Notice the difference in signs, which results directly from the order of subtraction. This difference highlights the "non-commutativity" of the operation.
Such awareness is crucial because assuming operations commute can lead to errors in calculations, particularly in higher mathematics and applied sciences where matrices frequently appear. Knowing when and why an operation doesn't commute ensures the accuracy and reliability of your mathematical work.