Chapter 7

Explain why a \(2 \times 2\) matrix will not have an inverse if either a column or a row contains all 0 s.

Perform each operation if possible. $$\left[\begin{array}{rr}12 & -5 \\ 10 & 3\end{array}\right]-\left[\begin{array}{rr}6 & 9 \\ -2 & 0\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 2 x-5 y+4 z &=-35 \\ 5 x+3 y-z &=1 \\ x+y+z &=1 \end{aligned}$$

Solve each system by substitution. $$\begin{aligned}-3 x+2 y &=-10 \\\9 x-6 y &=30\end{aligned}$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}0.4 & -0.8 & 0.6 \\\0.3 & 0.9 & 0.7 \\\3.1 & 4.1 & -2.8\end{array}\right]$$

Graph each inequality. $$y>(x-1)^{2}+2$$

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rr|r} 1 & -5 & 6 \\ 0 & 0 & 0 \end{array}\right]$$

Find the partial fraction decomposition for each rational expression. $$\frac{x^{3}+4}{9 x^{3}-4 x}$$

For each matrix, find \(A^{-1}\) if it exists.$$A=\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 1 & 0 & 1 \end{array}\right]$$

Perform each operation if possible. $$\left[\begin{array}{rrr}-8 & 4 & 0 \\ 2 & 5 & 0\end{array}\right]+\left[\begin{array}{ll}6 & 3 \\ 8 & 9\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} x-3 y-2 z &=-3 \\ 3 x+2 y-z &=12 \\ -x-y+4 z &=3 \end{aligned}$$

Solve each system by elimination. $$\begin{array}{l}x+y=5 \\\x-y=1\end{array}$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}-0.3 & -0.1 & 0.9 \\\2.5 & 4.9 & -3.2 \\\\-0.1 & 0.4 & 0.8\end{array}\right]$$

Graph each inequality. $$y>2(x+3)^{2}-1$$

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rr|r} 1 & 4 & -2 \\ 0 & 0 & 0 \end{array}\right]$$

Find the partial fraction decomposition for each rational expression. $$\frac{x^{3}+2}{x^{3}-3 x^{2}+2 x}$$

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{lll} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} x+y+z &=3 \\ 3 x-3 y-4 z &=-1 \\ x+y+3 z &=11 \end{aligned}$$

Solve each system by elimination. $$\begin{aligned}&x-y=-1\\\&x+y=-3\end{aligned}$$

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rrr|r} 1 & 1 & -1 & 4 \\ 0 & 1 & -1 & 2 \\ 0 & 0 & 1 & 1 \end{array}\right]$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}17 & -4 & 3 \\\11 & 5 & -15 \\\7 & -9 & 23\end{array}\right]$$

Graph each inequality. $$x^{2}+y^{2} \leq 4$$

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

Perform each operation if possible. $$\left[\begin{array}{rrrr}9 & 4 & 1 & -2 \\ 5 & -6 & 3 & 4 \\ 2 & -5 & 1 & 2\end{array}\right]-\left[\begin{array}{rrrr}-2 & 5 & 1 & 3 \\ 0 & 1 & 0 & 2 \\\ -8 & 3 & 2 & 1\end{array}\right]+\left[\begin{array}{rrrr}2 & 4 & 0 & 3 \\\ 4 & -5 & 1 & 6 \\ 2 & -3 & 0 & 8\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{array}{r} x-2 y+3 z=6 \\ 2 x-y+2 z=5 \end{array}$$

Solve each system by elimination. $$\begin{array}{r}x+2 y=6 \\\3 x-2 y=2\end{array}$$

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{rr}5 & x \\\\-3 & 2\end{array}\right]=6$$

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rrr|r} 1 & -2 & -1 & 0 \\ 0 & 1 & -3 & 1 \\ 0 & 0 & 1 & 2 \end{array}\right]$$

Graph each inequality. $$x^{2}+y^{2} \geq 1$$

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

Perform each operation if possible. $$\left[\begin{array}{rrr}6 & -2 & 4 \\ -2 & 5 & 8 \\ 1 & 0 & 2\end{array}\right]+\left[\begin{array}{rrr}3 & 0 & 8 \\ 1 & -2 & 4 \\ 6 & 9 & -2\end{array}\right]-\left[\begin{array}{rrr}-4 & 2 & 1 \\ 0 & 3 & -2 \\ 4 & 2 & 0\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 5 x-4 y+z &=9 \\ x+y\quad\quad &=15 \end{aligned}$$

Solve each system by elimination. $$\begin{array}{l}-x+4 y=0 \\\2 x-4 y=6\end{array}$$

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{rl}-0.5 & 2 \\\x & x\end{array}\right]=0$$

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rrr|r} 1 & 2 & -1 & 5 \\ 0 & 1 & -2 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Graph each inequality. $$x^{2}+(y+3)^{2} \leq 16$$

Find the partial fraction decomposition for each rational expression. $$\frac{3 x-2}{(x+4)\left(3 x^{2}+1\right)}$$

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{ccc} -0.4 & 0.1 & 0.2 \\ 0 & 0.6 & 0.8 \\ 0.3 & 0 & -0.2 \end{array}\right]$$

Perform each operation if possible. $$2\left[\begin{array}{rr}2 & -1 \\ 5 & 1 \\ 0 & 3\end{array}\right]+\left[\begin{array}{rr}5 & 0 \\ 7 & -3 \\ 1 & 1\end{array}\right]-\left[\begin{array}{rr}9 & -4 \\ 4 & 4 \\ 1 & 6\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 3 x+4 y-z &=13 \\ x+y+2 z &=15 \end{aligned}$$

Solve each system by elimination. $$\begin{array}{l}-x+3 y=8 \\\3 x+5 y=-10\end{array}$$

Solve each determinant equation for \(x\). $$\operatorname{det}\left[\begin{array}{ll}x & 3 \\\x & x\end{array}\right]=4$$

Each augmented matrix is in row echelon form and represents a linear system. Use back-substitution to solve the system if possible. $$\left[\begin{array}{rrr|r} 1 & -1 & 2 & 8 \\ 0 & 1 & -4 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Graph each inequality. $$(x-4)^{2}+y^{2} \leq 9$$

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

For each matrix, find \(A^{-1}\) if it exists. $$A=\left[\begin{array}{rrr} 0.8 & 0.2 & 0.1 \\ -0.2 & 0 & 0.3 \\ 0 & 0 & 0.5 \end{array}\right]$$

Perform each operation if possible. $$-3\left[\begin{array}{rr}3 & 8 \\ -1 & -9\end{array}\right]+5\left[\begin{array}{rr}4 & -8 \\ 1 & 6\end{array}\right]$$

Solve each system analytically. If the equations are dependent, write the solution set in terms of the variable \(z\). $$\begin{aligned} 3 x-5 y-4 z &=-7 \\ y-z &=-13 \end{aligned}$$

Solve each system by elimination. $$\begin{array}{l}3 x-7 y=18 \\\x+5 y=-16\end{array}$$

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