The following list of matrices and their respective characteristic polynomials is referred to in Exercise.

$$A=\left[\begin{array}{rr} 2 & -1 \\ -1 & 2 \end{array}\right]$$

, $p(t) = (t − 3)(t − 1),$

$$B=\left[\begin{array}{cc} 1 & -1 \\ 1 & 3 \end{array}\right]$$

, $p(t)=(t-2)^{2}$,

$$C=\left[\begin{array}{rrr} -6 & -1 & 2 \\ 3 & 2 & 0 \\ -14 & -2 & 5 \end{array}\right]$$

, $p(t)=-(t-1)^{2}(t+1)$,

$$D=\left[\begin{array}{rrr} -7 & 4 & -3 \\ 8 & -3 & 3 \\ 32 & -16 & 13 \end{array}\right]$$

, $p(t)=-(t-1)^{3}$,

$$E=\left[\begin{array}{llll} 6 & 4 & 4 & 1 \\ 4 & 6 & 1 & 4 \\ 4 & 1 & 6 & 4 \\ 1 & 4 & 4 & 6 \end{array}\right]$$

, $p(t) =(t + 1)(t + 5)^2(t − 15),$,

$$F=\left[\begin{array}{rrr} 1 & -1 & -1 & -1 \\ -1 & 1 & -1 & -1 \\ -1 & -1 & 1 & -1 \\ -1 & -1 & -1 & 1 \end{array}\right]$$

, $p(t)=(t+2)(t-2)^{3}$. Find a basis for the eigenspace $E_{\lambda}$ for the given matrix and the value of $\lambda$. Determine the algebraic and geometric multiplicities of $\lambda$. $E, \lambda=-1$

The following list of matrices and their respective characteristic polynomials is referred to in Exercise.

$$A=\left[\begin{array}{rr} 2 & -1 \\ -1 & 2 \end{array}\right]$$

, $p(t) = (t − 3)(t − 1),$

$$B=\left[\begin{array}{cc} 1 & -1 \\ 1 & 3 \end{array}\right]$$

, $p(t)=(t-2)^{2}$,

$$C=\left[\begin{array}{rrr} -6 & -1 & 2 \\ 3 & 2 & 0 \\ -14 & -2 & 5 \end{array}\right]$$

, $p(t)=-(t-1)^{2}(t+1)$,

$$D=\left[\begin{array}{rrr} -7 & 4 & -3 \\ 8 & -3 & 3 \\ 32 & -16 & 13 \end{array}\right]$$

, $p(t)=-(t-1)^{3}$,

$$E=\left[\begin{array}{llll} 6 & 4 & 4 & 1 \\ 4 & 6 & 1 & 4 \\ 4 & 1 & 6 & 4 \\ 1 & 4 & 4 & 6 \end{array}\right]$$

, $p(t) =(t + 1)(t + 5)^2(t − 15),$,

$$F=\left[\begin{array}{rrr} 1 & -1 & -1 & -1 \\ -1 & 1 & -1 & -1 \\ -1 & -1 & 1 & -1 \\ -1 & -1 & -1 & 1 \end{array}\right]$$

, $p(t)=(t+2)(t-2)^{3}$. Find a basis for the eigenspace $E_{\lambda}$ for the given matrix and the value of $\lambda$. Determine the algebraic and geometric multiplicities of $\lambda$. $E, \lambda=15$

Identify:

a) Hohokams,

b) Anasazi,

c) Mound Builders,

d) Inuits,

e) Iroquois League.

The Lincoln Saltdogs is a professional minor league baseball team in the American Association league. The clubhouse is insured for $600,000 under a commercial property insurance policy with an 80 percent coinsurance clause. The current replacement cost of the clubhouse is$1 million. After a playoff game for the league championship, a whirlpool tub for the players shorted out, and a fire ensued. The clubhouse sustained a $100,000 fire loss. Ignoring any deductible, how much will the team’s insurer pay for the loss?

Match the definitions with the words.

a. un article b. une pi$\`e$ce c. un po$\`e$me d. un roman e. un conte f. un journal g. un magazine h. une bande dessin$\'e$e

Balance the following chemical equations: (a) $\mathrm{CaCl}_{2}+\mathrm{Na}_{2} \mathrm{CO}_{3} \rightleftharpoons \mathrm{CaCO}_{3}+\mathrm{NaCl}$ (b) $\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}+\mathrm{O}_{2} \rightleftharpoons \mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O}$ (c) $\mathrm{NO}_{2}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{HNO}_{3}+\mathrm{NO}$ (d) $\mathrm{C}_{4} \mathrm{H}_{10}+\mathrm{O}_{2} \rightleftharpoons \mathrm{CO}_{2}+\mathrm{H}_{2} \mathrm{O}$ (e) $\mathrm{Al}(\mathrm{OH})_{2}^{-} \rightleftharpoons \mathrm{Al}^{3+}+\mathrm{OH}^{-}$

Assume that $X$ is a hypergeometric random variable with N=25, S=3, and n=4. Calculate the following probabilities.

a. P(X=0)

b. P(X=1)

c. P(X \leq 1)

Charlotte recorded the number of minutes she spent exercising in the past ten days: 12, 4, 5, 6, 8, 7, 9, 8, 2, 1. Find the median of the data.

The following reaction occurs when a burner on a gas stove is lit:

$$\mathrm{CH}_4(\mathrm{~g})+2 \mathrm{O}_2(\mathrm{~g})=\mathrm{CO}_2(\mathrm{~g})+2 \mathrm{H}_2 \mathrm{O}(\mathrm{g})$$

Is an equilibrium among $\mathrm{CH}_4, \mathrm{O}_2, \mathrm{CO}_2$, and $\mathrm{H}_2 \mathrm{O}$ established under these conditions? Explain your answer.

If $10.0\mathrm{~g}$ of sodium peroxide $\left(\mathrm{Na}_2 \mathrm{O}_2\right)$ react with water to produce sodium hydroxide and oxygen, how many liters of oxygen will be produced at $20 .{ }^{\circ} \mathrm{C}$ and $750 .\mathrm{torr}$?

$$2 \mathrm{~Na}_2 \mathrm{O}_2(s)+2 \mathrm{~H}_2 \mathrm{O}(l) \longrightarrow 4 \mathrm{~NaOH}(aq)+\mathrm{O}_2(g)$$

Oxygen $\left(\mathrm{O}_{2}\right)$ is heated during a steady-flow process at 1 atm from 298 to 3000 K at a rate of 0.5 kg/min. Determine the rate of heat supply needed during this process, assuming (a) some $\mathrm{O}_{2}$ dissociates into O and (b) no dissociation takes place.

The $z$-axis carries a constant electric charge density of $\lambda$ units of charge per unit length, with $\lambda > 0$. The resulting electric field is $\vec{E}$.

Compute the flux of $\vec{E}$ outward through the cylinder $x^2 + y^2 = R^2$, for $0 \leq z \leq h$.

Calculate the mass percent of solute in the following solutions.

a. 6.50 g of NaCl dissolved in 85.0 g of $\text{H}_2\text{O}$

b. 2.31 g of LiBr dissolved in 35.0 g of $\text{H}_2\text{O}$

c. 12.5 g of $\text{KNO}_3$ dissolved in 125 g of $\text{H}_2\text{O}$

d. 0.0032 g of NaOH dissolved in 1.2 g of $\text{H}_2\text{O}$

A radioactive substance ${R_{1}}$ having decay rate $k_{1}$ disintegrates into a second radioactive substance ${R_{2}}$ having decay rate ${k_{2} \neq k_{1}}$. Substance ${R_{2}}$ disintegrates into ${R_{3}}$, which is stable. If ${m_{i}(t)}$ represents the mass of substance ${R_{i}}$ at time ${t}$, ${i = 1, 2, 3}$, the applicable equations are

$$\begin{gather*} m'_{1} = −k_{1} m_{1} m'_{2} = k_{1} m_{1} − k_{2} m_{2} m'_{3} = k_{2} m_{2}. \end{gather*}$$

Use the Laplace transform to solve this system under the conditions

$$\begin{gather*} m_{1}(0) = m_{0}, m_{2}(0) = 0, m_{3}(0) = 0. \end{gather*}$$