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 .

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

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

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

Determine the dimension of the matrix.

$$\left[\begin{array}{rrrr}{3} & {-1} & {0} & {6} \\ {-2} & {7} & {1} & {4}\end{array}\right]$$

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

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

Use any method to solve the system.

$$\left\{\begin{aligned} 4 x-3 y =6 \\-5 x+7 y =-1 \end{aligned}\right.$$

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) $(B + C)A$ (b) $BA + CA$

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

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?

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

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 $A^{-1}$.

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

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

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

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.

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

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) ?

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

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

$$\left|\begin{array}{cc}{4 u} & {-1} \\ {-1} & {2 v}\end{array}\right|$$

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 + C), (b) AB + AC.