Problem 5 - College Algebra with Corequisite Support

Question

Does matrix multiplication commute? That is, does \(A B=B A ?\) If so, prove why it does. If not, explain why it does not.

Step-by-Step Solution

Answer

Matrix multiplication does not commute; generally, AB ≠ BA due to dependency on order.

Step 1: Understand Matrix Multiplication

Matrix multiplication involves performing dot products between rows of the first matrix (A) and columns of the second matrix (B). This operation results in a new matrix. Hence, for any two matrices A and B, to multiply them, the number of columns in A must be equal to the number of rows in B.

Step 2: Compare the Condition for Commutativity

For matrix multiplication to commute (i.e., AB = BA), both matrices A and B must be such that the operation can be performed in both orders. This requires that both A and B are square matrices of the same size, and even then, their commutativity is not guaranteed.

Step 3: Evaluate Matrix AB and Matrix BA

Calculate the product AB and then calculate the product BA. Observe that even if A and B are conformable for multiplication both ways, the resulting matrices AB and BA are not necessarily equal. This shows that matrix multiplication is dependent on the order of multiplication.

Step 4: Provide a Counterexample

Consider matrices A = \( \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \) and B = \( \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} \). Calculate AB and BA. AB = \( \begin{bmatrix} (1)(5)+(2)(7) & (1)(6)+(2)(8) \ (3)(5)+(4)(7) & (3)(6)+(4)(8) \end{bmatrix} = \begin{bmatrix} 19 & 22 \ 43 & 50 \end{bmatrix} \)BA = \( \begin{bmatrix} (5)(1)+(6)(3) & (5)(2)+(6)(4) \ (7)(1)+(8)(3) & (7)(2)+(8)(4) \end{bmatrix} = \begin{bmatrix} 23 & 34 \ 31 & 46 \end{bmatrix} \). These results differ, confirming that AB ≠ BA.

Step 5: Conclude with General Rule

Matrix commutation is not a general property of matrices. It is true only in special cases, such as when both matrices are diagonal or if one is the identity matrix. The general rule is that matrix multiplication does not commute.

Key Concepts

Commutative Property

The commutative property is a fundamental aspect of arithmetic, best known through simple operations like addition and multiplication of numbers. When we say an operation is commutative, it means that the order of the terms does not affect the result. For instance, for numbers:

• Addition: \(a + b = b + a\)
• Multiplication: \(a \times b = b \times a\)

However, when it comes to matrix multiplication, this property does not generally hold. Matrices usually do not commute; that is, for two matrices \(A\) and \(B\), the condition \(AB = BA\) is not often met. This is because matrix multiplication involves a series of calculations that inherently depend on the order in which the rows of the first matrix and the columns of the second matrix are considered.
Matrix multiplication's non-commutativity can lead to different outcomes when changing the multiplication order, making it a unique and significant contrast to basic numerical operations.

Square Matrices

Square matrices are matrices that have the same number of rows and columns. Understanding square matrices is crucial in discussing matrix commutation because they offer the only opportunity (albeit rare) for matrix multiplication to be commutative.
Consider a square matrix of size \(n\times n\). When you multiply two such matrices, they are always conformable, meaning the multiplication operation can proceed. However, this formality does not ensure the commutative property. Both matrices need to be square to even test for possible commutation, but most specific conditions, like one being a multiple of the other or both being diagonal, are also required for commutativity.
This unique arrangement allows some specialized operations like symmetry and orthogonality tests, which are vital in various mathematical and practical applications, including solving systems of equations and transformations in computer graphics.

Dot Product

The dot product (also called the scalar product) is a fundamental operation for matrix multiplication. Understanding how dot products work is key to working through matrix multiplication.

• For two vectors \(\mathbf{u} = [u_1, u_2, \ldots, u_n]\) and \(\mathbf{v} = [v_1, v_2, \ldots, v_n]\), the dot product is calculated as:

\[\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + \ldots + u_n v_n\]
This operation is similarly extended to matrices. In matrix multiplication, each element of the resulting matrix is the dot product of a row from the first matrix and a column from the second matrix. This step-wise calculation composes the new matrix.
The sequence of operations shows why the order matters: calculating \(AB\) involves the row elements of \(A\) interacting with column elements of \(B\), which is not symmetric if reversed to \(BA\). The essential point here is that because this process is ordered, and not interchangeable like scalar multiplication, commutation is often violated.

Counterexample

A counterexample is a simple yet powerful way to demonstrate a violation of some property or assumption. In mathematics, presenting a counterexample can clarify concepts and assumptions that might otherwise be obscured by more general or superficial observations.

Consider the matrices \(A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}\). When these are multiplied both ways:

• \(AB = \begin{bmatrix} 19 & 22 \ 43 & 50 \end{bmatrix}\)
• \(BA = \begin{bmatrix} 23 & 34 \ 31 & 46 \end{bmatrix}\)

The results are clearly different, illustrating that even for square matrices, multiplication does not necessarily commute. Counterexamples like this are useful to concretely exhibit why generalized statements about matrix properties must be made carefully. This understanding is especially important for those exploring linear algebra, ensuring clarity in foundational concepts before progressing further.