Chapter 6

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

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

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 3\begin{array}{l} 2 x+y+z=9 \\ -x-y+z=1 \\ 3 x-y+z=9 \end{array}

Find the partial fraction decomposition for each rational expression. $$\frac{4}{x(1-x)}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} 3 & 7 \\ 2 & 5 \end{array}\right]$$

Graph each inequality. Do not use a calculator. $$5 x \leq 4 y-2$$

Find the cofactor of each element in the second row for each matrix. $$\left[\begin{array}{rrr}-2 & 0 & 1 \\\1 & 2 & 0 \\\4 & 2 & 1\end{array}\right]$$

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises \begin{aligned} x+3 y+4 z &=14 \\ 2 x-3 y+2 z &=10 \\ 3 x-y+z &=9 \end{aligned}

Find the value of each variable. Do not use a calculator. $$\left[\begin{array}{ll} w & x \\ y & z \end{array}\right]=\left[\begin{array}{rr} 3 & 2 \\ -1 & 4 \end{array}\right]$$

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} x+5 y &=6 \\ x &=3 \end{aligned}$$

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

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} -5 & 3 \\ -8 & 5 \end{array}\right]$$

Graph each inequality. Do not use a calculator. $$2 x>3-4 y$$

Find the cofactor of each element in the second row for each matrix. $$\left[\begin{array}{rrr}1 & -1 & 2 \\\1 & 0 & 2 \\\0 & -3 & 1\end{array}\right]$$

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find \(x, y,\) and \(z\).) \begin{array}{r} 4 x-3 y+z=9 \\ 3 x+2 y-2 z=4 \\ x-y+3 z=5 \end{array}

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} 2 x+7 y &=1 \\ 5 x &=-15 \end{aligned}$$

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

Find the partial fraction decomposition for each rational expression. $$\frac{4 x^{2}-x-15}{x(x+1)(x-1)}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rr} -1 & -2 \\ 3 & 4 \end{array}\right]$$

Graph each inequality. Do not use a calculator. $$y<3 x^{2}+2$

Find the cofactor of each element in the second row for each matrix. $$\left[\begin{array}{rrr}1 & 2 & -1 \\\2 & 3 & -2 \\\\-1 & 4 & 1\end{array}\right]$$

Find the value of each variable. Do not use a calculator. $$\left[\begin{array}{rrc} 0 & 5 & x \\ -1 & 3 & y+2 \\ 4 & 1 & z \end{array}\right]=\left[\begin{array}{rrr} 0 & w+3 & 6 \\ -1 & 3 & 0 \\ 4 & 1 & 8 \end{array}\right]$$

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} 2 x+y+z &=3 \\ 3 x-4 y+2 z &=-7 \\ x+y+z &=2 \end{aligned}$$

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find $x, $$\begin{aligned} x+2 y+3 z &=8 \\ 3 x-y+2 z &=5 \\ -2 x-4 y-6 z &=5 \end{aligned}$$

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

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

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rr} -1 & 2 \\ -2 & -1 \end{array}\right]$$

Graph each inequality. Do not use a calculator. $$ y \leq x^{2}-4$$

If the equations are dependent, write the solution set in terms of the variable \(z\). (Hint: In Exercises 33-36, let \(t=\frac{1}{x}, u=\frac{1}{y},\) and \(v=\frac{1}{z} .\) Solve for \(t, u,\) and \(v,\) and then find $x, $$\begin{aligned} 3 x-2 y-8 z &=1 \\ 9 x-6 y-24 z &=-2 \\ x-y+z &=1 \end{aligned}$$

Find the cofactor of each element in the second row for each matrix. $$\left[\begin{array}{rrr}2 & -1 & 4 \\\3 & 0 & 1 \\\\-2 & 1 & 4\end{array}\right]$$

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} &4 x-2 y+3 z=4\\\ &3 x+5 y+z=7\\\ &5 x-y+4 z=7 \end{aligned}$$

Find the value of each variable. Do not use a calculator. $$\left[\begin{array}{ccc} 3+x & 4 & t \\ 5 & 8-w & y+1 \\ -4 & 3 & 2 r \end{array}\right]=\left[\begin{array}{ccc} 9 & 4 & 6 \\ z+3 & w & 9 \\ p & q & r \end{array}\right]$$

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

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

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} -6 & 4 \\ -3 & 2 \end{array}\right]$$

Graph each inequality. Do not use a calculator. $$y \leq 1-x^{2}$$

$$\begin{array}{r} x+4 y-z=6 \\ 2 x-y+z=3 \\ 3 x+2 y+3 z=16 \end{array}$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}4 & -7 & 8 \\\2 & 1 & 3 \\\\-6 & 3 & 0\end{array}\right]$$

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} x+y &=2 \\ 2 y+z &=-4 \\ z &=2 \end{aligned}$$

$$\left[\begin{array}{ccc} z & 4 r & 8 s \\ 6 p & 2 & 5 \end{array}\right]+\left[\begin{array}{ccc} -9 & 8 r & 3 \\ 2 & 5 & 4 \end{array}\right]=\left[\begin{array}{ccc} 2 & 36 & 27 \\ 20 & 7 & 12 a \end{array}\right]$$

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

Find the partial fraction decomposition for each rational expression. $$\frac{3}{x^{2}+4 x+3}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rr} 5 & 10 \\ -3 & -6 \end{array}\right]$$

Graph each inequality. Do not use a calculator. $$y<2-3 x^{2}$$

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}8 & -2 & -4 \\\7 & 0 & 3 \\\5 & -1 & 2\end{array}\right]$$

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} x &=6 \\ y+2 z &=2 \\ x-3 z &=6 \end{aligned}$$

Find the value of each variable. Do not use a calculator. $$\left[\begin{array}{ccc} a+2 & 1 & 5 m \\ 8 k & 0 & 3 \end{array}\right]+\left[\begin{array}{ccc} 3 a & 2 z & 5 m \\ 2 k & 5 & 6 \end{array}\right]=\left[\begin{array}{ccc} 10 & -14 & 80 \\ 10 & 5 & 9 \end{array}\right]$$

Solve each system by substitution. $$\begin{aligned}&y=2 x+3\\\&3 x+4 y=78\end{aligned}$$

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{ll} 0.6 & 0.2 \\ 0.5 & 0.1 \end{array}\right]$$

Graph each inequality. Do not use a calculator. $$y>(x-1)^{2}+2$$