How would you describe her actions after the mirror cracks? Do you think she could have responded differently? Explain your answer.

For each case, express $U$ as the product of elementary matrices (without calculation) and use this factorization to find $U^{-1}$.

(a)

$$U=\left[\begin{array}{rrr}1 & 2 & 0 \\ 0 & 1 & -5 \\ 0 & 0 & 1\end{array}\right]$$

(b)

$$U=\left[\begin{array}{rrr}1 & -1 & 9 \\ 0 & 1 & 4 \\ 0 & 0 & 1\end{array}\right]$$

(c)

$$U=\left[\begin{array}{lll}1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1\end{array}\right]~~~$$

(d)

$$U=\left[\begin{array}{rrrr}1 & -2 & -4 & 3 \\ 0 & 1 & -7 & -5 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 0 & 1\end{array}\right]$$

.

We know that F(x)=^x∫0 e^e^t dt is a continuous function by FTC1, though it is not an elementary function. The functions ∫ e^x/x dx and ∫ 1/ln x dx are not elementary either, but they can be expressed in terms of E. Evaluate the following integrals in terms of F. ^3∫2 1/ln x dx

See the problem statement preceding Prob. and use $T=5 \mathrm{kN} \cdot \mathrm{m}$.

The molecules $\mathrm{BF}_{3}, \mathrm{CF}_{4}, \mathrm{CO}_{2}, \mathrm{PF}_{5}, \text { and } \mathrm{SF}_{6}$ are all nonpolar, even though they contain polar bonds. Why?

Express each of the following as a product of elementary matrices (if possible), in the manner of the said example:

(a)

$$\left[\begin{array}{ll}4 & 9 \\ 3 & 7\end{array}\right]\qquad$$

(c)

$$\left[\begin{array}{rrrr}0 & 0 & 5 & 0 \\ -3 & 0 & 0 & -2 \\ 0 & 6 & -10 & -1 \\ 3 & 0 & 0 & 3\end{array}\right]$$

(b)

$$\left[\begin{array}{rrr} -3 & 2 & 1 \\ 13 & -8 & -9 \\ 1 & -1 & 2 \end{array}\right]$$

find elementary matrices $E_{1}, E_{2},$ and $E_3$ such that $A E_{1} E_{2} E_{3}=L$, where L is lower triangular. Calculate the product $Q=E_{1} E_{2} E_{3}$ and verify that $AQ = L$ and $\operatorname{det}(Q)=\operatorname{det}\left(Q^{T}\right)$.

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

Fill the table by writing each given equation in its alternate form.

See the problem statement preceding Prob. , and use $T=5 \mathrm{kN} \cdot \mathrm{m}$.

Use elementary row or column operations to find the determinant.

$$\begin{vmatrix} 3&-2&4&3&1\\-1&0&2&1&0\\5&-1&0&3&2\\4&7&-8&0&0 \\1&2&3&0&2\end{vmatrix}$$

In the preceding problem, how far downward (from its original trajectory) is the electron deflected during the time it's in the electric field?

Use elementary row operations together with the Cofactor Expansion Theorem to evaluate the given determinant.

$$\left| \begin{array} { r r r r } { 3 } & { 5 } & { 2 } & { 6 } \\ { 2 } & { 3 } & { 5 } & { - 5 } \\ { 7 } & { 5 } & { - 3 } & { - 16 } \\ { 9 } & { - 6 } & { 27 } & { - 12 } \end{array} \right|.$$

Find the elementary divisors of the given group:

$$\text { (a) } \mathbb{Z}_{250}$$

$$\text { (b) } \mathbb{Z}_{6} \oplus \mathbb{Z}_{12} \oplus \mathbb{Z}_{18}$$

$$\text { (c) } \mathbb{Z}_{10} \oplus \mathbb{Z}_{20} \oplus \mathbb{Z}_{30} \oplus \mathbb{Z}_{40}$$

$$\text { (d) } \mathbb{Z}_{12} \oplus \mathbb{Z}_{30} \oplus \mathbb{Z}_{100} \oplus \mathbb{Z}_{240}$$

Solve the problem.

The focus of the main parabolic mirror of a telephoto Iens is $10$ in. above the vertex. A hyperbolic mirror is placed so that its vertex is at $(0,8)$. If the hyperbola has center $(0,0)$ and foci $(0, \pm 10)$, then what is the equation of the hyperbola?