Chapter 7

Graph each inequality. $$x \leq 3$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}-5 & 9 \\\4 & -1\end{array}\right]$$

Determine whether A and B are imerses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{ll} 5 & 7 \\ 2 & 3 \end{array}\right] ; B=\left[\begin{array}{rr} 3 & -7 \\ -2 & 5 \end{array}\right]$$

Find the dimension of each matrix. Identify any square, column, or rove matrices. Do not use a calculator. $$\left[\begin{array}{rr}-3 & 6 \\\7 & -4\end{array}\right]$$

Verify that the given ordered triple is a solution of the system. Do not use a calculator. $$\begin{aligned} (-3,6,1) & \\ 2 x+y-z=&-1 \\ x-y+3 z=&-6 \\ -4 x+y+z=& 19 \end{aligned}$$

Graph each inequality. $$y \leq-2$$

Determine whether A and B are imerses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{ll} 2 & 3 \\ 1 & 1 \end{array}\right] ; B=\left[\begin{array}{rr} -1 & 3 \\ 1 & -2 \end{array}\right]$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{ll}-1 & 3 \\\\-2 & 9\end{array}\right]$$

Find the dimension of each matrix. Identify any square, column, or rove matrices. Do not use a calculator. $$\left[\begin{array}{rrr}2 & -8 & 6 \\ 1 & 0 & -5 \\ 5 & -2 & 3\end{array}\right]$$

Verify that the given ordered triple is a solution of the system. Do not use a calculator. $$\begin{aligned} \left(\frac{1}{2},-\frac{3}{4}, \frac{1}{6}\right) & \\ 2 x+8 y-6 z &=-6 \\ x+y+z &=-\frac{1}{12} \\ x+3 z &=1 \end{aligned}$$

Graph each inequality. $$y>1$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}-1 & -2 \\\5 & 3\end{array}\right]$$

Determine whether A and B are imerses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{rr} -1 & 2 \\ 3 & -5 \end{array}\right] ; B=\left[\begin{array}{ll} -5 & -2 \\ -3 & -1 \end{array}\right]$$

Find the dimension of each matrix. Identify any square, column, or rove matrices. Do not use a calculator. $$\left[\begin{array}{rrrr}-6 & 8 & 0 & 0 \\ 4 & 1 & 9 & 2 \\ 3 & -5 & 7 & 1\end{array}\right]$$

Verify that the given ordered triple is a solution of the system. Do not use a calculator. $$\begin{aligned} (-0.2,0.4,0.5) & \\ 5 x-y+2 z &=-0.4 \\ x+4 z &=1.8 \\ -3 y+z &=-0.7 \end{aligned}$$

Graph each inequality. $$x<-2$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{ll}6 & -4 \\\0 & -1\end{array}\right]$$

Determine whether the partial fraction decomposition of the first expression is the second expression. See Example 1. $$\frac{3 x-1}{x^{2}-x} ; \frac{1}{x}+\frac{3}{x-1}$$

Find the dimension of each matrix. Identify any square, column, or rove matrices. Do not use a calculator. $$\left[\begin{array}{rrrr}-3 & 4 & 2 & 1 \\ 0 & 8 & 6 & 3\end{array}\right]$$

Verify that the given ordered triple is a solution of the system. Do not use a calculator. $$\begin{aligned} (-1,-2,-3) & \\ x-y+z=&-2 \\ x-2 y+z=& 0 \\ y-z=& 1 \end{aligned}$$

Graph each inequality. $$y \geq x+1$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}9 & 3 \\\\-3 & -1\end{array}\right]$$

Determine whether the partial fraction decomposition of the first expression is the second expression. See Example 1. $$\frac{1}{x^{2}(x-1)} ; \frac{1}{x-1}-\frac{1}{x}-\frac{1}{x^{2}}$$

Determine whether A and B are imerses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{rrr} 0 & 1 & 0 \\ 0 & 0 & -2 \\ 1 & -1 & 0 \end{array}\right] ; B=\left[\begin{array}{rrr} 1 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & -1 & 0 \end{array}\right]$$

Find the dimension of each matrix. Identify any square, column, or rove matrices. Do not use a calculator. $$\left[\begin{array}{l}2 \\ 4\end{array}\right]$$

Verify that the given ordered triple is a solution of the system. Do not use a calculator. $$\begin{aligned} (-2,-1,3) & \\ x-y+z &=2 \\ 3 x-2 y+z &=-1 \\ x+y &=-3 \end{aligned}$$

Solve each system by substitution $$\begin{aligned}6 x-y &=5 \\\y &=x\end{aligned}$$.

Graph each inequality. $$y \geq x-2$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{ll}0 & 2 \\\1 & 5\end{array}\right]$$

Determine whether A and B are imerses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{lll} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \end{array}\right] ; B=\left[\begin{array}{lll} 1 & -2 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{array}\right]$$

Find the dimension of each matrix. Identify any square, column, or rove matrices. Do not use a calculator. $$\left[\begin{array}{ll}4 & 9\end{array}\right]$$

Verify that the given ordered triple is a solution of the system. Do not use a calculator. $$\begin{aligned} &\left(\frac{1}{2}, \frac{1}{2},-2\right)\\\ &3 x+y+z=0\\\ &4 x+2 y+z=1\\\ &2 x-2 y-z=2 \end{aligned}$$

Solve each system by substitution. $$\begin{aligned}5 x+y &=2 \\\y &=-3 x\end{aligned}$$

Graph each inequality. $$y \leq-2 x$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}3 & 4 \\\5 & -2\end{array}\right]$$

Find the partial fraction decomposition for each rational expression. $$\frac{5}{3 x(2 x+1)}$$

Determine whether A and B are imerses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{rrr} -1 & -1 & -1 \\ 4 & 5 & 0 \\ 0 & 1 & -3 \end{array}\right] ; B=\left[\begin{array}{rrr} 15 & 4 & -5 \\ -12 & -3 & 4 \\ -4 & -1 & 1 \end{array}\right]$$

Write the augmented matrix for each system. Do not solve the system. $$\begin{array}{r} 2 x+3 y=11 \\ x+2 y=8 \end{array}$$

Find the dimension of each matrix. Identify any square, column, or rove matrices. Do not use a calculator. $$[-9]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} x+y+z &=5 \\ y+z &=2 \\ y-z &=0 \end{aligned}$$

Solve each system by substitution. $$\begin{aligned}x+2 y &=-1 \\\2 x+y &=4\end{aligned}$$

Graph each inequality. $$y \leq \frac{1}{2} x$$

Find each determinant. Do not use a calculator. $$\operatorname{det}\left[\begin{array}{rr}-9 & 7 \\\2 & 6\end{array}\right]$$

Find the partial fraction decomposition for each rational expression. $$\frac{3 x-1}{x(x+1)}$$

Determine whether A and B are imerses by calculating AB and BA. Do not use a calculator. $$A=\left[\begin{array}{lll} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{array}\right] ; B=\left[\begin{array}{rrr} 7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{array}\right]$$

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} &3 x+5 y=-13\\\ &2 x+3 y=-9 \end{aligned}$$

Find the dimension of each matrix. Identify any square, column, or rove matrices. Do not use a calculator. $$\left[\begin{array}{lllll}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} x-y-z &=6 \\ y+z &=-3 \\ y-z &=1 \end{aligned}$$

Solve each system by substitution. $$\begin{array}{r}2 x+y=-11 \\\x+3 y=-8\end{array}$$

Graph each inequality. $$x+y>1$$