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Chapter 2: Problem 26

Show that \(A B\) and \(B A\) are not equal for the given matrices. $$A=\left[\begin{array}{ll}\frac{1}{4} & \frac{1}{2} \\\\\frac{1}{2} & \frac{1}{2}\end{array}\right], \quad B=\left[\begin{array}{ll}\frac{1}{2} & \frac{1}{2} \\\\\frac{1}{2} & \frac{1}{4}\end{array}\right]$$

Short Answer

Expert verified
After performing the matrix multiplication for A*B and B*A, it can be observed that the resulting matrices are not equal. Hence, for the given matrices A and B, A*B and B*A are not equal.

Step by step solution

01

Matrix Multiplication A*B

The first step is to multiply matrix A with B. This is done using the standard method of matrix multiplication: the element in each position of the resulting matrix is achieved by multiplying the elements of the corresponding row in the first matrix by the elements of the corresponding column in the second matrix and summing these products together. So, for A*B, \(1/4*1/2+1/2*1/2=0.375\) for the first row, first column element, and so on for the rest of the elements.
02

Matrix Multiplication B*A

In the next step, multiply matrix B with A. Although it might seem similar to the first step, the change in order of multiplication changes the rows and columns to be multiplied together. Hence the resulting matrix will be different. For example, for B*A, \(1/2*1/4+1/2*1/2=0.375\) for the first row, first column element, and so on.
03

Comparing the Results

The last step is to compare the two matrices resulting from steps 1 and 2. If any element in the same position differs in two matrices, it can be concluded that the matrices are not equal, thereby proving that AB 鈮� BA.

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Key Concepts

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

Linear Algebra

Linear algebra is a foundational field of mathematics that deals with vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties that are common to all vector spaces. A key operation in linear algebra is matrix multiplication, which is used to map between different vector spaces, solve systems of linear equations, and transform geometric objects.

Matrices, which are organized as rectangular arrays of numbers, are not only tools for computations but also carry deeper theoretical significance. A key concept in linear algebra is that matrix multiplication can represent a composite of linear transformations. This implies that applying one linear transformation after another is equivalent to applying a single transformation represented by the product of their corresponding matrices.

Non-Commutative Matrices

Matrix multiplication is a non-commutative operation in most cases. The term 'non-commutative' means that the order in which two matrices are multiplied affects the result of the multiplication. In other words, given two matrices, A and B, the product of AB is typically not the same as the product of BA.

This non-commutative nature of matrix multiplication is fundamental to the study of linear algebra, as it implies that the sequence in which linear transformations are applied is important. Consequently, understanding the properties of matrix multiplication is essential for applications in various fields, such as physics, computer graphics, and engineering where matrix transformations often represent critical operations.

Matrix Operations

A basic matrix operation is matrix multiplication, where matrices of appropriate dimensions can be combined to produce a new matrix. The process involves taking rows from the first matrix and columns from the second matrix, multiplying the corresponding elements, and summing these products to form the elements of the new matrix. This operation is used not only to represent the composition of linear transformations but also as a tool in solving systems of linear equations and analyzing complex systems.

Unlike addition and multiplication of real numbers, which are commutative (the order does not affect the result), with matrices, the order of multiplication matters. To indicate that matrix multiplication is not commutative, it's frequently taught through exercises showing the different results obtained by multiplying the same matrices in a different order, which reflects the essence of linear transformations within different coordinate contexts.

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Most popular questions from this chapter

Use elementary matrices to find the inverse of $$ \begin{array}{l} A=\left[\begin{array}{lll} 1 & a & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\left[\begin{array}{lll} 1 & 0 & 0 \\ b & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & c \end{array}\right] \end{array} $$

Find a sequence of elementary matrices whose product is the given nonsingular matrix. $$ \left[\begin{array}{ll} 1 & 1 \\ 2 & 1 \end{array}\right] $$

Is the sum of two invertible matrices invertible? Explain why or why not. Illustrate your conclusion with appropriate examples.

Verify that \((A B)^{T}=B^{T} A^{T}\). $$ A=\left[\begin{array}{rrr}2 & 1 & -1 \\\0 & 1 & 3 \\\4 & 0 & 2\end{array}\right] \text { and } B=\left[\begin{array}{rrr}1 & 0 & -1 \\ 2 & 1 & -2 \\\0 & 1 & 3\end{array}\right] $$

In a population of \(10,000,\) there are 5000 nonsmokers, 2500 smokers of one pack or less per day, and 2500 smokers of more than one pack per day. During any month, there is a \(5 \%\) probability that a nonsmoker will begin smoking a pack or less per day, and a \(2 \%\) probability that a nonsmoker will begin smoking more than a pack per day. For smokers who smoke a pack or less per day, there is a \(10 \%\) probability of quitting and a \(10 \%\) probability of increasing to more than a pack per day. For smokers who smoke more than a pack per day, there is a \(5 \%\) probability of quitting and a \(10 \%\) probability of dropping to a pack or less per day. How many people will be in each group (a) in 1 month, (b) in 2 months, and \((\mathrm{c})\) in 1 year?
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