Problem 76 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the statement is true or false. Justify your answer. $$ \left[\begin{array}{rr} -6 & -2 \\ 2 & -6 \end{array}\right]\left[\begin{array}{rr} 4 & 0 \\ 0 & -1 \end{array}\right]=\left[\begin{array}{rr} 4 & 0 \\ 0 & -1 \end{array}\right]\left[\begin{array}{rr} -6 & -2 \\ 2 & -6 \end{array}\right] $$

Step-by-Step Solution

Answer

Matrix multiplication, in general, is not commutative. Thus, AB is not always equal to BA. To justify the truth of the statement given, we need to calculate AB and BA and then compare the resulting matrices. The results will show whether the given statement is true or false.

Step 1: Compute AB

Multiply matrix A with matrix B using the rule of matrix multiplication. To find the element in the i-th row and j-th column of the resultant matrix, take the dot product of the i-th row of A and the j-th column of B.

Step 2: Compute BA

Similarly, compute BA by taking the dot product of the i-th row of B and the j-th column of A.

Step 3: Compare AB and BA

The statement is true if and only if every corresponding entry in AB and BA are equal. So, compare the resultant matrices obtained in Steps 1 and 2.

Key Concepts

Understanding Linear Algebra

At the heart of many scientific and engineering disciplines lies a powerful mathematical toolbox known as linear algebra. It is the study of vectors, vector spaces, and linear mappings between such spaces. In this field, matrices play a crucial role as they can represent linear transformations or systems of linear equations.

In practice, matrices are arrays of numbers or functions arranged in rows and columns that can be manipulated through operations such as addition, subtraction, and, importantly, multiplication. Linear algebra equips us with methods for solving these complex systems, crucial for fields such as computer graphics, optimization, quantum mechanics, and even the study of networks.

Demystifying the Dot Product

When multiplying matrices, we use a specific process involving the dot product, also known as the scalar product. This operation takes two equal-length sequences of numbers and returns a single number. Here's how it works: you multiply corresponding entries and then sum up those products.

To get a clearer image, consider two vectors, \( \textbf{a} = (a_1, a_2) \) and \( \textbf{b} = (b_1, b_2) \). Their dot product is \( a_1b_1 + a_2b_2 \). In matrix multiplication, we use the dot product to calculate each entry of the resulting matrix by dot-multiplying the corresponding row of the first matrix and the column of the second matrix.

• Understand that the length of rows and columns directly affects the ability to multiply two matrices.
• Remember that the dot product is a critical step in matrix multiplication, not an optional one.

Is Matrix Multiplication Commutative?

A common misconception students might have is thinking that the commutative property of matrices works the same way as it does for real numbers. However, this is not the case. In general, matrix multiplication is not commutative; that is to say, for two matrices A and B, it's typically false that \( AB = BA \).

The exercise provided is a perfect example of this, asking us to verify whether two matrices – when multiplied in different orders – yield the same product. In reality, the multiplication of matrices often results in two different matrices unless the matrices hold certain properties, like being diagonal and of the same dimensions or being identity matrices. This peculiar aspect of matrix behavior is a fundamental concept in linear algebra and is essential for students to understand to prevent confusion when solving equations or transforming spaces.