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Chapter 2: Problem 27

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.

Short Answer

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#tag_title#Step 2: Compute B*A#tag_content#Now, let's compute the product of B and A, or B*A. Here is the multiplication process: \[ B*A = \left[\begin{array}{cc} 2 & 1 \\ 4 & 3 \end{array}\right] * \left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right] = \left[\begin{array}{cc} (2*1 + 1*3) & (2*2 + 1*4) \\ (4*1 + 3*3) & (4*2 + 3*4) \end{array}\right] \] #tag_title#Step 3: Compare A*B and B*A#tag_content#Now compare the results of A*B and B*A: \( A*B = \left[\begin{array}{cc} 10 & 7 \\ 22 & 15 \end{array}\right] \) \( B*A = \left[\begin{array}{cc} 5 & 10 \\ 13 & 24 \end{array}\right] \) Since A*B 鈮� B*A, we can conclude that matrix multiplication is not commutative in general.

Step by step solution

01

Compute A*B

To compute the product of A and B, or A*B, we need to multiply the elements in the rows of A by the elements in the columns of B. Here is the multiplication process: \[ A*B = \left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right] * \left[\begin{array}{cc} 2 & 1 \\ 4 & 3 \end{array}\right] = \left[\begin{array}{cc} (1*2 + 2*4) & (1*1 + 2*3) \\ (3*2 + 4*4) & (3*1 + 4*3) \end{array}\right] \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Non-Commutative Property in Matrix Multiplication
In mathematics, the non-commutative property refers to the behavior of certain operations for which the order of the operands affects the outcome of the operation. This is notably distinct from commutative operations, like addition and multiplication of real numbers, where the order does not change the result.

When dealing with matrix multiplication, this non-commutative property is prominently observed. Unlike multiplying regular numbers, where the equation, for example, 2 times 3 is always equal to 3 times 2, matrix multiplication does not follow this rule. This means that for two matrices, A and B, the product AB is generally not equal to the product BA. The exercise provided perfectly illustrates this phenomenon. The matrices A and B can be multiplied in both orders, resulting in AB and BA, but the product matrices are typically not the same.

Understanding this property is crucial in linear algebra as it impacts how we can manipulate and use matrices in various applications like physics, computer graphics, and engineering.
Matrices Operations
Matrix operations are fundamental to linear algebra, encompassing various procedures such as addition, subtraction, multiplication, and the finding of inverses. Among these, matrix multiplication is a particularly important operation and is more complex than it might initially appear.

As shown in the exercise, to multiply two matrices A and B to find AB, one must take the row elements from the first matrix, A, and multiply them by the corresponding column elements of the second matrix, B. We then add up the products to find each element of the resulting matrix. This technique is consistent regardless of the size of the matrices involved, although they must be compatible to be multiplied (the number of columns in the first matrix must equal the number of rows in the second).

Matrix multiplication is used in many applications, such as transforming shapes in computer graphics, solving systems of linear equations, and modeling complex systems in various scientific disciplines. Mastery of this operation is essential for any student of linear algebra.
Linear Algebra
Linear algebra is a branch of mathematics that is primarily concerned with vectors, vector spaces, and linear mappings between these spaces. It is a foundational topic in applied and theoretical mathematics, as well as in physics and engineering.

Linear algebra deals with several mathematical structures and transformations that are used in everyday computations and in the understanding of geometric concepts. It provides the language to describe lines, planes, and subspaces in ways that allow for solving equations and predicting outcomes in a variety of contexts, from the simplest to the most complex.

The concepts of linear algebra are used to solve systems of linear equations, to describe geometric transformations, and to model situations in which things change constistently, like the motion of a car or the growth of a population. With its widespread application in computer science, quantum mechanics, and operations research, having a thorough understanding of linear algebra is highly beneficial for students across scientific and engineering disciplines.

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Most popular questions from this chapter

Mr. and Mrs. Garcia have a total of $$\$ 100,000$$ to be invested in stocks, bonds, and a money market account. The stocks have a rate of return of \(12 \% /\) year, while the bonds and the money market account pay \(8 \% /\) year and \(4 \%\) year, respectively. The Garcias have stipulated that the amount invested in the money market account should be equal to the sum of \(20 \%\) of the amount invested in stocks and \(10 \%\) of the amount invested in bonds. How should the Garcias allocate their resources if they require an annual income of $$\$$ 10,000$ from their investments?

The amount of money raised by charity I, charity II, and charity III (in millions of dollars) in each of the years 2006,2007, and 2008 is represented by the matrix \(A\) : $$ A=\begin{array}{cccc} & \mathrm{I} & \mathrm{II} & \mathrm{III} \\ 2006 & {\left[\begin{array}{ccc} 18.2 & 28.2 & 40.5 \\ 19.6 & 28.6 & 42.6 \\ 20.8 & 30.4 & 46.4 \end{array}\right]} \\ 2007 & 2008 \end{array} $$ On average, charity I puts \(78 \%\) toward program cost, charity II puts \(88 \%\) toward program cost, and charity III puts \(80 \%\) toward program cost. Write a \(3 \times 1\) matrix \(B\) reflecting the percentage put toward program cost by the charities. Then use matrix multiplication to find the total amount of money put toward program cost in each of the 3 yr by the charities under consideration.

Let $$ A=\left[\begin{array}{rr} 1 & 3 \\ -2 & -1 \end{array}\right] \text { and } B=\left[\begin{array}{ll} 3 & -4 \\ 2 & -2 \end{array}\right] $$ a. Find \(A^{T}\) and show that \(\left(A^{T}\right)^{T}=A\). b. Show that \((A+B)^{T}=A^{T}+B^{T}\). c. Show that \((A B)^{T}=B^{T} A^{T}\).

Three network consultants, Alan, Maria, and Steven, each received a year-end bonus of \(\$ 10,000\), which they decided to invest in a \(401(\mathrm{k})\) retirement plan sponsored by their employer. Under this plan, employees are allowed to place their investments in three funds: an equity index fund (I), a growth fund (II), and a global equity fund (III). The allocations of the investments (in dollars) of the three employees at the beginning of the year are summarized in the matrix $$ \begin{array}{l} \text { II }\\\ \begin{array}{c} \text { Alan } \\ A=\text { Maria } \\ \text { Steven } \end{array}\left[\begin{array}{lll} 4000 & 3000 & 3000 \\ 2000 & 5000 & 3000 \\ 2000 & 3000 & 5000 \end{array}\right] \end{array} $$ The returns of the three funds after 1 yr are given in the matrix $$ \begin{array}{r} \mathrm{I} \\ B=\mathrm{II} \\ \mathrm{III} \end{array}\left[\begin{array}{l} 0.18 \\ 0.24 \\ 0.12 \end{array}\right] $$ Which employee realized the best return on his or her investment for the year in question? The worst return?

The problems in exercise correspond to those in exercises 15-27, Section 2.1. Use the results of your previous work to help you solve these problems. The annual returns on Sid Carrington's three investments amounted to $$\$ 21,600$$: \(6 \%\) on a savings account, \(8 \%\) on mutual funds, and \(12 \%\) on bonds. The amount of Sid's investment in bonds was twice the amount of his investment in the savings account, and the interest earned from his investment in bonds was equal to the dividends he received from his investment in mutual funds. Find how much money he placed in each type of investment.
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