In given exercise, the matrix product AX performs the translation of the point (x, y) to the point (x+h, y+k), when

$$A=\left[\begin{array}{lll}1 & 0 & h \\0 & 1 & k \\0 & 0 & 1\end{array}\right] \text { and } X=\left[\begin{array}{l}x \\y \\1\end{array}\right] \text {. }$$

Predict the coordinates of the translated point.

$$A=\left[\begin{array}{rrr}1 & 0 & -3 \\ 0 & 1 & 4 \\ 0 & 0 & 1\end{array}\right], \quad X=\left[\begin{array}{r}2 \\ -4 \\ 1\end{array}\right]$$

Use a graphing utility to perform the operations for the matrices A, B, and C and the scalar c. Write a brief statement comparing the results of parts (a) and (b).

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

,

$$B=\left[\begin{array}{rrr}{2} & {3} & {0} \\ {4} & {1} & {-2} \\ {-1} & {2} & {0}\end{array}\right]$$

,

$$C=\left[\begin{array}{rrr}{3} & {-2} & {1} \\ {-4} & {0} & {3} \\ {-1} & {3} & {-2}\end{array}\right]$$

, and c = 3. (a) $(A+B)^{2}$ (b) $A^{2}+A B+B A+B^{2}$

Solve the system graphically or algebraically. Explain your choice of method.

$$\left\{\begin{aligned} x y-1 &=0 \\ -5 x-2 y+1 &=0 \end{aligned}\right.$$

In given exercise find a system of line a equations that has the given solution. (There are many correct answers.)

(0,8)

In given exercise determine the order of the matrix.

[14]

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $\frac{1}{x^{2}+x}$

In given exercise, the matrix product A X performs the translation of the point (x, y) to the point (x+h, y+k), when

$$A=\left[\begin{array}{lll} 1 & 0 & h \\ 0 & 1 & k \\ 0 & 0 & 1 \end{array}\right] \text { and } X=\left[\begin{array}{c} x \\ y \\ 1\end{array}\right]$$

Find $A^{-1} B$. What does $A^{-1} B$ represent?

$$A=\left[\begin{array}{rrr}1 & 0 & -2 \\ 0 & 1 & 3 \\ 0 & 0 & 1\end{array}\right], \quad X=\left[\begin{array}{r}1 \\ -2 \\ 1\end{array}\right]$$

In given exercise, the matrix product A X performs the translation of the point (x, y) to the point (x+h, y+k), when

$$A=\left[\begin{array}{lll} 1 & 0 & h \\ 0 & 1 & k \\ 0 & 0 & 1 \end{array}\right] \text { and } X=\left[\begin{array}{c} x \\ y \\ 1\end{array}\right]$$

Find $A^{-1}$ .

$$A=\left[\begin{array}{rrr}1 & 0 & -2 \\ 0 & 1 & 3 \\ 0 & 0 & 1\end{array}\right], \quad X=\left[\begin{array}{r}1 \\ -2 \\ 1\end{array}\right]$$

In given exercise, the matrix product A X performs the translation of the point (x, y) to the point (x+h, y+k), when

$$A=\left[\begin{array}{lll} 1 & 0 & h \\ 0 & 1 & k \\ 0 & 0 & 1 \end{array}\right] \text { and } X=\left[\begin{array}{c} x \\ y \\ 1\end{array}\right]$$

Find B=A X .

$$A=\left[\begin{array}{rrr}1 & 0 & -2 \\ 0 & 1 & 3 \\ 0 & 0 & 1\end{array}\right], \quad X=\left[\begin{array}{r}1 \\ -2 \\ 1\end{array}\right]$$

In given exercise, the matrix product A X performs the translation of the point (x, y) to the point (x+h, y+k), when

$$A=\left[\begin{array}{lll} 1 & 0 & h \\ 0 & 1 & k \\ 0 & 0 & 1 \end{array}\right] \text { and } X=\left[\begin{array}{c} x \\ y \\ 1\end{array}\right]$$

Predict the coordinates of the translated point

$$A=\left[\begin{array}{rrr}1 & 0 & -2 \\ 0 & 1 & 3 \\ 0 & 0 & 1\end{array}\right], \quad X=\left[\begin{array}{r}1 \\ -2 \\ 1\end{array}\right]$$

In given exercise, the matrix product A X performs the translation of the point (x, y) to the point (x+h, y+k), when

$$A=\left[\begin{array}{lll} 1 & 0 & h \\ 0 & 1 & k \\ 0 & 0 & 1 \end{array}\right] \text { and } X=\left[\begin{array}{c} x \\ y \\ 1\end{array}\right]$$

What are the coordinates of the point (x, y) ?

$$A=\left[\begin{array}{rrr}1 & 0 & -2 \\ 0 & 1 & 3 \\ 0 & 0 & 1\end{array}\right], \quad X=\left[\begin{array}{r}1 \\ -2 \\ 1\end{array}\right]$$

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus.

$$\left|\begin{array}{cc}{3 x^{2}} & {-3 y^{2}} \\ {1} & {1}\end{array}\right|$$

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $\frac{6 x+5}{(x+2)^{4}}$

Use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system.

$$\left\{\begin{array}{r} x+y+z=0 \\ 2 x+3 y+z=0 \\ 3 x+5 y+z=0 \end{array}\right.$$