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

(0,8)

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

$$\left|\begin{array}{ll}{x} & {\ln x} \\ {1} & {1 / x}\end{array}\right|$$

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 $A^{-1}$ .

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

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

$$\left\{\begin{array}{r} 2 \ln x+y=4 \\ e^{x}-y=0 \end{array}\right.$$

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 .

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

Determine whether the statement is true or false. Justify your answer. When the product of two square matrices is the identity matrix, the matrices are inverses of one another.

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) ?

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

Use the inverse of the coefficient matrix of this system to find the amount invested in each type of bond.

Total Investment 500000

Annual Return 8000

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) $BA + CA$

In given exercise, use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system.

$$\left\{\begin{aligned} 3 x+3 y+12 z & =6 \\ x+y+4 z & =2 \\ 2 x+5 y+20 z & =10 \\ -x+2 y+8 z & =4\end{aligned}\right.$$

Use any method to solve the system.

$$\left\{\begin{aligned} -5 x+9 y &=13 \\ y &=x-4 \end{aligned}\right.$$

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

$$\left|\begin{array}{cc}{e^{-x}} & {x e^{-x}} \\ {-e^{-x}} & {(1-x) e^{-x}}\end{array}\right|$$