Reduce $[A | I]$ to find $A^{−1}$. In each case, check your calculations by multiplying the given matrix by the derived inverse.

$$\left[\begin{array}{lll} 1 & 3 & 5 \\ 0 & 1 & 4 \\ 0 & 2 & 7 \end{array}\right]$$

A 0.72-mol sample of $\mathrm{PCl}_{5}$ is put into a 1.00-L vessel and heated. At equilibrium, the vessel contains 0.40 mol of $\mathrm{PCl}_{3}$(g) and 0.40 mol of $\mathrm{Cl}_{2}$(g). Calculate the value of the equilibrium constant for the decomposition of $\mathrm{PCl}_{5}$ to $\mathrm{PCl}_{3}$and $\mathrm{Cl}_{2}$ at this temperature.

If the pressure reading of your pitot tube is 15.0 mm Hg at a speed of 200 km/h, what will it be at 700 km/h at the same altitude?

Use Eq. $V \mathbf{x}=\theta$ to determine whether the given set of vectors is linearly independent or linearly dependent.

$$\mathbf{v}_{1}=\left[\begin{array}{l} 1 \\ 2 \end{array}\right], \ \mathbf{v}_{2}=\left[\begin{array}{l} 2 \\ 3 \end{array}\right], \ \mathbf{v}_{3}=\left[\begin{array}{l} 2 \\ 4 \end{array}\right], \ \mathbf{v}_{4}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], \ \mathbf{v}_{5}=\left[\begin{array}{l} 3 \\ 6 \end{array}\right]$$

,

$$\mathbf{u}_{0}=\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right], \ \mathbf{u}_{1}=\left[\begin{array}{r} 1 \\ 2 \\ -1 \end{array}\right], \ \mathbf{u}_{2}=\left[\begin{array}{r} 2 \\ 1 \\ -3 \end{array}\right], \ \mathbf{u}_{3}=\left[\begin{array}{r} -1 \\ 4 \\ 3 \end{array}\right], \ \mathbf{u}_{4}=\left[\begin{array}{l} 4 \\ 4 \\ 0 \end{array}\right], \ \mathbf{u}_{5}=\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right]$$

If the set is linearly dependent, express one vector in the set as a linear combination of the others. $\left\{\mathbf{u}_{1}, \mathbf{u}_{4}, \mathbf{u}_{5}\right\}$

A 0.0500-kg ice cube at $-30.0^{\circ} \mathrm{C}$ is placed in 0.400 kg of $35.0^{\circ} \mathrm{C}$ water in a very well-insulated container. What is the final temperature?

Find the function f given that the stone of the tangent line to the graph at any point (x, f(x)) is f'(x) and that the graph of passes through the given point. $f^{\prime}(x)=\frac{x}{\sqrt{x^{2}+1}} ;(0,1)$

Write Lewis structures and IUPAC names for the alkyne isomers of $\mathrm{C}_{4} \mathrm{H}_{6}$.

Marbles dropped into a partially filled bathtub sink to the bottom. Part of their weight is supported by buoyant force, yet the downward force on the bottom of the tub increases by exactly the weight of the marbles. Explain why.

(a) A 75.0-kg man floats in freshwater with 3.00% of his volume above water when his lungs are empty, and 5.00% of his volume above water when his lungs are full. Calculate the volume of air he inhales-called his lung capacity-in liters. (b) Does this lung volume seem reasonable?

The life span (in years) of a certain make of car battery is an exponentially distributed random variable with an expected value of 5. Find the probability that the life span of a battery is (a) less than 4 years, (b) more than 6 years, and (c) between 2 and 4 years.

Why does a dull hypodermic needle hurt more than a sharp one?

Consider the system

$$\begin{array}{l} x^{\prime}=-3 x, \\ y^{\prime}=-2 x-y, \\ z^{\prime}=-2 z. \end{array}$$

(a) Find the eigenvalues and eigenvectors. Use your numerical solver to sketch the half-lines generated by the exponential solutions $y(t)=C_{i} e^{\lambda_{i} t} v_{i}$ for i=1, 2, and 3. (b) The six half-line solutions found in part (a) come in pairs that form straight lines. These three lines, taken two at a lime, generate three planes. In turn these planes divide phase space into eight octants. Use your numerical solver to add solution trajectories with initial conditions in each of the eight octants. (c) Based on your phase portrait, what would be an appropriate name for the equilibrium point in this case?

Use Eq. $V \mathbf{x}=\theta$ to determine whether the given set of vectors is linearly independent or linearly dependent.

$$\mathbf{v}_{1}=\left[\begin{array}{l} 1 \\ 2 \end{array}\right], \ \mathbf{v}_{2}=\left[\begin{array}{l} 2 \\ 3 \end{array}\right], \ \mathbf{v}_{3}=\left[\begin{array}{l} 2 \\ 4 \end{array}\right], \ \mathbf{v}_{4}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right], \ \mathbf{v}_{5}=\left[\begin{array}{l} 3 \\ 6 \end{array}\right]$$

,

$$\mathbf{u}_{0}=\left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right], \ \mathbf{u}_{1}=\left[\begin{array}{r} 1 \\ 2 \\ -1 \end{array}\right], \ \mathbf{u}_{2}=\left[\begin{array}{r} 2 \\ 1 \\ -3 \end{array}\right], \ \mathbf{u}_{3}=\left[\begin{array}{r} -1 \\ 4 \\ 3 \end{array}\right], \ \mathbf{u}_{4}=\left[\begin{array}{l} 4 \\ 4 \\ 0 \end{array}\right], \ \mathbf{u}_{5}=\left[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right]$$

If the set is linearly dependent, express one vector in the set as a linear combination of the others. $\left\{\mathbf{v}_{2}, \mathbf{v}_{3}, \mathbf{v}_{4}\right\}$

Consider a person climbing and descending stairs. Construct a problem in which you calculate the long-term rate at which stairs can be climbed considering the mass of the person, his ability to generate power with his legs, and the height of a single stair step. Also consider why the same person can descend stairs at a faster rate for a nearly unlimited time in spite of the fact that very similar forces are exerted going down as going up. (This points to a fundamentally different process for descending versus climbing stairs.)