91Ó°ÊÓ

Write the uncoded row matrices for the message. Then encode the message using the matrix \(A\). Message: Row Matrix Size: Encoding Matrix: SELL CONSOLIDATED \(1 \times 3\) \(A=\left[\begin{array}{rrr}1 & -1 & 0 \\ 1 & 0 & -1 \\ -6 & 2 & 3\end{array}\right]\)

\(A\) medical researcher is studying the spread of a virus in a population of 1000 laboratory mice. During any week, there is an \(80 \%\) probability that an infected mouse will overcome the virus, and during the same week there is a \(10 \%\) probability that a noninfected mouse will become infected. Three hundred mice are currently infected with the virus. How many will be infected (a) next week and (b) in 3 weeks?

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?

Determine whether the stochastic matrix \(P\) is regular. Then find the steady state matrix \(X\) of the Markov chain with matrix of transition probabilities \(P\). $$P=\left[\begin{array}{ll} 0.2 & 0 \\ 0.8 & 1 \end{array}\right]$$

Find the matrix product \(A B C\) by (a) grouping the factors as \((A B) C,\) and \((b)\) grouping the factors as \(A(B C) .\) Show that you obtain the same result from both processes. $$\begin{aligned}&A=\left[\begin{array}{rr}-4 & 2 \\\1 & -3\end{array}\right], \quad B=\left[\begin{array}{rrr}1 & -5 & 0 \\\\-2 & 3 & 3\end{array}\right]\\\&C=\left[\begin{array}{rr}-3 & 4 \\\0 & 1 \\\\-1 & 1\end{array}\right]\end{aligned}$$

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

Wildlife \(A\) wildlife management team studied the reproduction rates of deer in three tracts of a wildlife preserve. The team recorded the number of females \(x\) in each tract and the percent of females \(y\) in each tract that had offspring the following year. The table shows the results. $$ \begin{array}{l|ccc} \hline \text {Number, }, x & 100 & 120 & 140 \\ \text {Percent, } y & 75 & 68 & 55 \\ \hline \end{array} $$ (a) Find the least squares regression line for the data. (b) Use a graphing utility to graph the model and the data in the same viewing window. (c) Use the model to create a table of estimated values for y. Compare the estimated values with the actual data. (d) Use the model to estimate the percent of females that had offspring when there were 170 females. (e) Use the model to estimate the number of females when \(40 \%\) of the females had offspring.

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

Explain why the formula is not valid for matrices. Illustrate your argument with examples. $$ (A+B)(A-B)=A^{2}-B^{2} $$

Determine whether the Markov chain with matrix of transition probabilities \(P\) is absorbing. Explain. $$P=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 0.3 & 0.9 \\ 0 & 0.7 & 0.1 \end{array}\right]$$