(a) Give a counterexample to show that $$ ( A B ) ^ { - 1 } \neqA ^ { - 1 } B ^ { - 1 } $$ in general. (h) Under what conditions on A and B is $$ ( A B ) ^ { - 1 } =A ^ { - 1 } B ^ { - 1 } ? $$ Prove your assertion.

A 1-m^3 tank contains 2.841 kg of steam at 0.6 MPa. Determine the temperature of the steam, using (a) the ideal-gas equation, (b) the van der Waals equation, and (c) the steam tables. Answers: (a) 457.6 K, (b) 465.9 K, (c) 473 K

Suppose $A$ and $B$ are sets, and $f$ and $g$ are functions with $f:A\rightarrow B$ and $g:B\rightarrow A$. Prove: If $g\circ f=\text{id}_A$ and $f\circ g=\text{id}_B$, then $f$ is invertible and $g=f^{-1}$. Note: This result is a converse to Proposition 26.9.

Solve each system for x and y expressing either value in terms of a or b, if necessary. Assume that

$$a \neq 0 \text { and } b \neq 0.$$

For the linear function

$$f ( x ) = m x + b , f ( - 2 ) = 11 \text { and } f ( 3 ) = - 9. Find m and b.$$

Determine the decimal value of each signed binary number in the 2’s complement form: (a) 10011001 (b) 01110100 (c) 10111111

Does ( 4 , - 1 ) satisfy x - 2 y = 6 ?

Use expanded forms to convert the given base two numeral to base ten.

$$10101_{ t w o}$$

$$\left. \begin{array} { l } { \text { What are the orders of the elements of } D _ { 15 } ? \text { How many elements } } \\ { \text { have each of these orders? } } \end{array} \right.$$

a. Prove that $\mathbf{I}^{2}=\mathbf{I}$ for any identity matrix $\mathbf{I}.$ b. Prove that $\mathbf{I}^{n}=\mathbf{I}$ for any identity matrix $\mathbf{I}$ and any positive integer n.

Prove that, for square matrices A and B, AB = BA if and only if

$$( A - B ) ( A + B ) = A ^ { 2 } - B ^ { 2 }.$$

Sketch the graph of each equation.

$$y = \frac { 3 } { 4 } x - 1$$

Find the indefinite integral.

$$∫ \frac{e^x}{(e^x - 1)(e^x + 4)} dx$$

Describe the ways in which mRNA can be modified to code for different proteins:

If A and B are

$$n \times n$$

matrices, prove the following properties of the trace: (a) tr (A + B) = tr (A) + tr (B) (b) tr (kA) = ktr (A), where k is a scalar