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 whether each ordered pair is a solution of the system of equations.

$$\left\{\begin{array}{l}{4 x-y=1} \\ {6 x+y=-6}\end{array}\right.$$

(a) $(0, −3)$ (b) $(−1, −5)$ (c) $\left(-\frac{3}{2}, 3\right)$ (d) $\left(-\frac{1}{2},-3\right)$

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

(0,8)

Let matrices A, B, C, and D be of dimensions 2 x 3, 2 x 3, 3 x 2, and 2 x 2, respectively. Determine whether the matrices are of proper dimension to perform the operation(s). If so, give the dimension of the answer. $C D$

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?

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

Is a consistent system with exactly one solution independent or dependent?

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 general, when multiplying matrices A and B, does AB = BA?

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) $A^{2}+A B+B A+B^{2}$

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.

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

Is a consistent system with exactly one solution independent or dependent?

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) ?

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

Let A be a 3 x 3 matrix such that $|A|=5$. Can you use this information to find $|2 A|$? Explain.