Verify the distributive $\mid$ law $A$ ($B$+$C$)= $AB$+$AC$ when

$$\mathbf{A}=\left(\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rrrr} 2 & -1 & 1 & 0 \\ 3 & -1 & 2 & 1 \end{array}\right), \quad \mathbf{C}=\left(\begin{array}{rrrr} -1 & 1 & 1 & 2 \\ -2 & 2 & 0 & -1 \end{array}\right)$$

Compute $\mathbf{a}+\mathbf{b}, \mathbf{a}-\mathbf{b}, 2 \mathbf{a}+3 \mathbf{b}$, and $-5 \mathbf{a}+2 \mathbf{b}$ when

$$\mathbf{a}=\left(\begin{array}{r}2 \\ -1\end{array}\right)$$

and

$$\mathbf{b}=\left(\begin{array}{l}3 \\ 4\end{array}\right)$$

.

Find the transposes of

$$\mathbf{A}=\left(\begin{array}{rrrr}3 & 5 & 8 & 3 \\ -1 & 2 & 6 & 2\end{array}\right), \mathbf{B}=\left(\begin{array}{r}0 \\ 1 \\ -1 \\ 2\end{array}\right), \mathbf{C}=(1,5,0,-1)$$

.

Decide which of the following single equations in the variables $x, y, z$, and $w$ are linear and which are not. (In (f), $a$ and $b$ are nonnegative constants.)

(a) $3 x-y-z-w=50$

(b) $\sqrt{3} x+8 x y-z+w=0$

(c) $3(x+y-z)=4(x-2 y+3 z)$

(d) $3.33 x-4 y+\frac{800}{3} z=3$

(e) $(x-y)^2+3 z-w=-3$

(f) $2 a^2 x-\sqrt{b} y+(2+\sqrt{a}) z=b^2$

Construct the matrix $A$ =$\left(a_{i j}\right)_{3 \times 3}$, where $a_{i i}=1$ for $i=1,2,3$, and $a_{i j}=0$ for $i \neq j$.

Find the equation for the line:

(a) that passes through points $(3,-2,2)$ and $(10,2,1)$.

(b) that passes through point $(1,3,2)$ and has the same direction as $(0,-1,1)$.

Solve the following systems by Gaussian elimination.

(a) $x_1+x_2=3$

$$3 x_1+5 x_2=5$$

(b) $x_1+2 x_2+x_3=4$

$$x_1-x_2+x_3=5$$

$$2 x_1+3 x_2-x_3=1$$

(c) $2 x_1-3 x_2+x_3=0$

$$x_1+x_2-x_3=0$$

$$2 x_1+3 x_2-x_3=1$$

You ride a bicycle. In what sense is your bicycle solar powered?

Let

$$\begin{aligned} & A=\left[\begin{array}{rrr} 1 & 0 & -2 \\ 1 & -3 & 2 \\ -2 & 1 & 1 \end{array}\right] \quad B=\left[\begin{array}{rrr} 3 & 1 & 0 \\ 2 & 2 & 0 \\ 1 & -3 & -1 \end{array}\right] \\ & C=\left[\begin{array}{lll} 2 & -1 & 0 \\ 1 & -1 & 2 \\ 3 & -2 & 1 \end{array}\right] \end{aligned}$$

Verify the validity of the distributive law for matrix multiplication.

Why is Earth's Moon unique among all moons in the solar system?

Cadmium telluride is an important material for solar cells. What wavelength of light would a photon of this energy correspond to?

Let

$$A = \left[ \begin{array} { l l } { 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right] , \quad B = \left[ \begin{array} { r r } { 1 } & { 0 } \\ { 0 } & { - 1 } \end{array} \right] , \quad C = \left[ \begin{array} { r r } { - 1 } & { 0 } \\ { 0 } & { 1 } \end{array} \right] \\D = \left[ \begin{array} { r r } { - 1 } & { 0 } \\ { 0 } & { - 1 } \end{array} \right].$$

Use any three of the matrices to verify a distributive property.

A certain source gives 2000 counts per second at time $t=0$. Its half-life is $2 \mathrm{~min}$. (a) What is the counting rate after $4 \mathrm{~min}$ ? (b) After $6 \mathrm{~min}$ ? (c) After $8 \mathrm{~min}$ ?

Kennedy's teacher asked her to plan the budget for the class party. Kennedy began by wnting the expression g = b + 7 to represent that the number of girls equa)s the number of boys plus seven. Eacli girl will need $6. Write and simplify an algebraic expression that uses the Distributive Property to show the total cost for girls at the class party.