Rationalize the following lattice energy values:

$$\begin{matrix} \text{Compound} & & \text{Lattice Energy (kJ/mol)}\\ \hline \\ \begin{array} { l } { \text { CaSe } } \\ { \mathrm { Na } _ { 2 } \mathrm { Se } } \\ { \text { CaTe } } \\ { \mathrm { Na } _ { 2 } \mathrm { Te } } \end{array} & & \begin{array} { l } { - 2862 } \\ { - 2130 } \\ { - 2721 } \\ { - 2095 } \end{array} \end{matrix}$$

Assume that the $(3 \times 3)$ matrix A satisfies det(A) = 2, where A is given by

$$A=\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right]$$

Calculate det(B) in each case.

$$B=\left[\begin{array}{lll} d & f & e \\ a & c & b \\ g & i & h \end{array}\right]$$

Some rows from Pascal's triangle are shown. What is the next row in the given case?

$$\begin{array}{llllllllll}1 & 9 & 36 & 84 & 126 & 126 & 84 & 36 & 9 & 1\end{array}$$

Write the matrix with the given elements. In each case, specify the dimensions of the matrix.

$$\left. \begin{array} { l } { a _ { 11 } = 2 , a _ { 12 } = 1 , a _ { 13 } = - 1 , a _ { 21 } = 0 , a _ { 22 } = 4 }, \\ { a _ { 23 } = - 2 . } \end{array} \right.$$

Graph each case-defined function and give the domain and range.

$$g(x)=\left\{\begin{aligned} x & \text { if } 0 \leq x<1 \\ x^{2}-2 x+2 & \text { if } x \geq 1 \end{aligned}\right.$$

A table of values for a linear function is given. Fill in the missing value and calculate m in each case.

$$\begin{matrix} \text{x} & \text{-1} & \text{0} & \text{1}\\ \text{y} & \text{5} & \text{8}\\ \end{matrix}$$

The given is a function $f: \mathrm{R} \rightarrow \mathrm{R}$. Determine (a) whether or not f is injective, and (b) whether or not f is surjective. Prove your answer in either case.

$$f(x)=\left\{\begin{aligned} x & \text { if } x \text { is rational } \\ 2 x & \text { if } x \text { is irrational }\end{aligned}\right.$$

Determine the elgenvalues and corresponding elgenvectors of

$$A=\left[\begin{array}{cc}2 & 3-3 t \\ 3+3 i & 5\end{array}\right]$$

.

Verify that eigenvectors corresponding to different eigenvalues are orthogonal. Why should this be the case?

Assume that the $(3 \times 3)$ matrix A satisfies det(A) = 2, where A is given by

$$A=\left[\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right]$$

Calculate det(B) in each case.

$$B=\left[\begin{array}{lll} b & a & c \\ e & d & f \\ h & g & i \end{array}\right]$$

Solve each system of equations without graphing. For each case, explain what the solution tells you about the graph of the system. a.

$$\begin{array}{l}{y=\frac{1}{3} x^{2}+1} \\ {y=2 x-2}\end{array}$$

, b.

$$\begin{array}{l}{x=y^{2}} \\ {x-y=6}\end{array}$$

, c.

$$\begin{array}{l}{6 x-2 y=-4} \\ {y=3 x+2}\end{array}$$

.

Verify that $W$ is a subspace of $V.$ In each case, assume that $V$ has the standard operations.

$W$ is the set of all $3 \times 2$ matrices of the form

$$\left[ \begin{array}{rr}{a} & {b} \\ {a-2 b} & {0} \\ {0} & {c}\end{array}\right].$$

$$V=M_{3,2}$$

Graph each case-defined function and give the domain and range.

$$\gamma(x)=\left\{\begin{aligned} 3 x & \text { if } 0 \leq x<2 \\ 10-x^{2} & \text { if } x \geq 2 \end{aligned}\right.$$

W is a subspace of the vector space V of all $(2 \times 2)$ matrices. A matrix A in W is written as

$$A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$$

In each case exhibit a basis for W. $W=\{A: b=a-c, \ d=2 a+c\}$

A table of values for a linear function is given. Fill in the missing value and calculate m in each case.

$$\begin{matrix} \text{x} & \text{2} & \text{4} & \text{5}\\ \text{f(x)} & \text{-1} & \text{-2}\\ \end{matrix}$$