91Ó°ÊÓ

determine whether the matrix is elementary. If it is, state the elementary row operation used to produce it. $$\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]$$

determine whether the matrix is elementary. If it is, state the elementary row operation used to produce it. $$\left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]$$

The market research department at a manufacturing plant determines that \(20 \%\) of the people who purchase the plant's product during any month will not purchase it the next month. On the other hand, \(30 \%\) of the people who do not purchase the product during any month will purchase it the next month. In a population of 1000 people, 100 people purchased the product this month. How many will purchase the product next month? In 2 months?

A population of 10,000 is grouped as follows: 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 of the 3 groups in 1 month? In 2 months?

Factor the matrix \(A\) into a product of elementary matrices. $$A=\left[\begin{array}{ll} 1 & 2 \\ 1 & 0 \end{array}\right]$$

Factor the matrix \(A\) into a product of elementary matrices. $$A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 2 & 5 & 6 \\ 1 & 3 & 4 \end{array}\right]$$

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) The zero matrix is an elementary matrix. (b) A square matrix is nonsingular if it can be written as the product of elementary matrices. (c) \(A \mathbf{x}=O\) has only the trivial solution if and only if \(A \mathbf{x}=\mathbf{b}\) has a unique solution for every \(n \times 1\) column matrix b.

Is the product of two elementary matrices always elementary? Explain why or why not and provide appropriate examples to illustrate your conclusion.

Is the sum of two elementary matrices always elementary? Explain why or why not and provide appropriate examples to illustrate your conclusion.

show that the matrix is invertible and find its inverse. $$A=\left[\begin{array}{rr} \sin \theta & \cos \theta \\ -\cos \theta & \sin \theta \end{array}\right]$$