Problem 27 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let $$A=\left[\begin{array}{ll}1 & 2 \\\3 & 4 \end{array}\right] \text { and } B=\left[\begin{array}{ll}2 & 1 \\\4 & 3\end{array}\right]$$ Compute \(A B\) and \(B A\) and hence deduce that matrix multiplication is, in general, not commutative.

Step-by-Step Solution

Answer

We computed the products of the given matrices A and B and found that \(AB = \left[\begin{array}{ll}10 & 7 \\\22 & 15\end{array}\right]\) and \(BA = \left[\begin{array}{ll}5 & 8 \\\13 & 20\end{array}\right]\). As the results are different, we can deduce that matrix multiplication is not commutative in general.

Step 1: Compute AB

To compute the product \(AB\), multiply each element of the ith row of \(A\) with the corresponding element of the jth column of \(B\) and then sum the results. Let's find the elements of matrix \(AB = \left[\begin{array}{ll}a & b \\\c & d\end{array}\right]\): \(a = (1)(2) + (2)(4) = 2 + 8 = 10\) \(b = (1)(1) + (2)(3) = 1 + 6 = 7\) \(c = (3)(2) + (4)(4) = 6 + 16 = 22\) \(d = (3)(1) + (4)(3) = 3 + 12 = 15\) So, the matrix product \(AB = \left[\begin{array}{ll}10 & 7 \\\22 & 15\end{array}\right]\).

Step 2: Compute BA

To compute the product \(BA\), multiply each element of the ith row of \(B\) with the corresponding element of the jth column of \(A\) and then sum the results. Let's find the elements of matrix \(BA = \left[\begin{array}{ll}e & f \\\g & h\end{array}\right]\): \(e = (2)(1) + (1)(3) = 2 + 3 = 5\) \(f = (2)(2) + (1)(4) = 4 + 4 = 8\) \(g = (4)(1) + (3)(3) = 4 + 9 = 13\) \(h = (4)(2) + (3)(4) = 8 + 12 = 20\) So, the matrix product \(BA = \left[\begin{array}{ll}5 & 8 \\\13 & 20\end{array}\right]\).

Step 3: Compare AB and BA

Now, let's compare the resulting matrices \(AB\) and \(BA\): \(AB = \left[\begin{array}{ll}10 & 7 \\\22 & 15\end{array}\right]\) \(BA = \left[\begin{array}{ll}5 & 8 \\\13 & 20\end{array}\right]\) We can see that the matrices \(AB\) and \(BA\) are not the same. Since the order of multiplication changes the result, we can deduce that matrix multiplication is not commutative in general.

Key Concepts

Non-Commutative Property

Matrix multiplication displays a fundamental property that often surprises those new to linear algebra: it is typically non-commutative. This means that, in general, the order in which you multiply matrices affects the result.
When you have two matrices, say \(A\) and \(B\), the product \(AB\) is usually different from \(BA\). This is unlike arithmetic with real numbers where, for example, \(2 \times 3 = 3 \times 2\).

Consider matrices \(A\) and \(B\) from our exercise. We calculate \(AB\) and \(BA\) and find:

• \(AB = \begin{bmatrix} 10 & 7 \ 22 & 15 \end{bmatrix}\)
• \(BA = \begin{bmatrix} 5 & 8 \ 13 & 20 \end{bmatrix}\)

Clearly, \(AB eq BA\). This example illustrates that matrix multiplication does not adhere to the commutative property. This non-commutative nature is crucial for various applications of matrices in mathematics and physics, where the direction or sequence of operations directly affects outcomes.

Matrix Product Calculation

Calculating the product of two matrices involves a methodical process. This process involves aligning rows from the first matrix with columns from the second.
For instance, consider the product \(AB\). To calculate each element of this product:

• Take the first row of \(A\) and the first column of \(B\), multiply corresponding elements, and then sum them to get the first element \(a\) of the resulting matrix.
• Repeat for each row of \(A\) with each column of \(B\) to fill out the whole matrix.

In our exercise, we found:
\(a = (1)(2) + (2)(4) = 10\)
\(b = (1)(1) + (2)(3) = 7\)
Similarly for other elements.
Remember, this new matrix's dimensions will be determined by the number of rows of the first matrix and columns of the second. Thus, you must ensure the inner dimensions match (columns of the first matrix equals rows of the second) to perform multiplication.

Linear Algebra

Linear algebra is a cornerstone of mathematics dealing with vectors, matrices, and linear transformations.
Matrices, rectangular arrangements of numbers, encode information and operations that relate to transformations in a plane or space. They generalize many mathematical concepts and provide tools for solving systems of linear equations.

In the context of linear algebra, understanding matrix operations such as addition, multiplication, and their properties is essential.

• Matrix addition is straightforward and commutative.
• Matrix multiplication, however, as explored in our exercise, is more complex and non-commutative.

Applications of this subject appear in diverse fields like physics for rotational transformations, computer graphics for scaling and rotating images, and economics for input-output models. Linear algebra provides powerful methods to solve problems involving large data sets, understanding dynamic systems and optimizing solutions.