Chapter 10

Write the form of the partial fraction decomposition of the function (as in Example 4). Do not determine the numerical values of the coefficients. \(\frac{x^{3}-4 x^{2}+2}{\left(x^{2}+1\right)\left(x^{2}+2\right)}\)

\(1-8\) Find the determinant of the matrix, if it exists. $$ \left[\begin{array}{cc}{\frac{1}{2}} & {\frac{1}{8}} \\ {1} & {\frac{1}{2}}\end{array}\right] $$

Find the inverse of the matrix if it exists. \(\left[\begin{array}{ll}{5} & {3} \\ {3} & {2}\end{array}\right]\)

7–14 A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$\left[\begin{array}{rrr}{1} & {0} & {-3} \\ {0} & {1} & {5}\end{array}\right]$$

Use back-substitution to solve the triangular system. $$ \left\\{\begin{aligned} x+2 y+z &=7 \\\\-y+3 z &=9 \\ 2 z &=6 \end{aligned}\right. $$

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{aligned} x+y &=4 \\\\-x+y &=0 \end{aligned}\right.$$

Use the substitution method to find all solutions of the system of equations. \(\left\\{\begin{aligned} x+y^{2} &=0 \\ 2 x+5 y^{2} &=75 \end{aligned}\right.\)

1–14 Graph the inequality. $$3 x+4 y+12>0$$

Write the form of the partial fraction decomposition of the function (as in Example 4). Do not determine the numerical values of the coefficients. \(\frac{x^{4}+x^{2}+1}{x^{2}\left(x^{2}+4\right)^{2}}\)

\(1-8\) Find the determinant of the matrix, if it exists. $$ \left[\begin{array}{rr}{2.2} & {-1.4} \\ {0.5} & {1.0}\end{array}\right] $$

Find the inverse of the matrix if it exists. \(\left[\begin{array}{ll}{3} & {4} \\ {7} & {9}\end{array}\right]\)

Perform the matrix operation, or if it is impossible, explain why. $$ \left[\begin{array}{lll}{2} & {1} & {2} \\ {6} & {3} & {4}\end{array}\right]\left[\begin{array}{rr}{1} & {-2} \\ {3} & {6} \\ {-2} & {0}\end{array}\right] $$

7–14 A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$\left[\begin{array}{rrr}{1} & {3} & {-3} \\ {0} & {1} & {5}\end{array}\right]$$

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{array}{l}{x-y=3} \\ {x+3 y=7}\end{array}\right.$$

Use back-substitution to solve the triangular system. $$ \left\\{\begin{aligned} x-2 y+3 z &=10 \\ 2 y-z &=2 \\ 3 z &=12 \end{aligned}\right. $$

Use the substitution method to find all solutions of the system of equations. \(\left\\{\begin{aligned} x^{2}-y &=1 \\ 2 x^{2}+3 y &=17 \end{aligned}\right.\)

1–14 Graph the inequality. $$4 x+5 y<20$$

Write the form of the partial fraction decomposition of the function (as in Example 4). Do not determine the numerical values of the coefficients. \(\frac{x^{3}+x+1}{x(2 x-5)^{3}\left(x^{2}+2 x+5\right)^{2}}\)

\(9-14\) Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{11}, A_{11} $$

Find the inverse of the matrix if it exists. \(\left[\begin{array}{rr}{2} & {5} \\ {-5} & {-13}\end{array}\right]\)

Perform the matrix operation, or if it is impossible, explain why. $$ \left[\begin{array}{rr}{1} & {2} \\ {-1} & {4}\end{array}\right]\left[\begin{array}{rrr}{1} & {-2} & {3} \\ {2} & {2} & {-1}\end{array}\right] $$

7–14 A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$\left[\begin{array}{llll}{1} & {2} & {8} & {0} \\ {0} & {1} & {3} & {2} \\\ {0} & {0} & {0} & {0}\end{array}\right]$$

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{array}{l}{2 x-3 y=9} \\ {4 x+3 y=9}\end{array}\right.$$

Use back-substitution to solve the triangular system. $$ \left\\{\begin{aligned} 2 x-y+6 z &=5 \\ y+4 z &=0 \\\\-2 z &=1 \end{aligned}\right. $$

Use the elimination method to find all solutions of the system of equations. \(\left\\{\begin{array}{r}{x+2 y=5} \\ {2 x+3 y=8}\end{array}\right.\)

1–14 Graph the inequality. $$-x^{2}+y \geq 10$$

Write the form of the partial fraction decomposition of the function (as in Example 4). Do not determine the numerical values of the coefficients. \(\frac{1}{\left(x^{3}-1\right)\left(x^{2}-1\right)}\)

\(9-14\) Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{33}, A_{33} $$

Find the inverse of the matrix if it exists. \(\left[\begin{array}{rr}{-7} & {4} \\ {8} & {-5}\end{array}\right]\)

Perform the matrix operation, or if it is impossible, explain why. $$ \left[\begin{array}{rr}{2} & {-3} \\ {0} & {1} \\ {1} & {2}\end{array}\right]\left[\begin{array}{l}{5} \\ {1}\end{array}\right] $$

7–14 A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$\left[\begin{array}{rrrr}{1} & {0} & {-7} & {0} \\ {0} & {1} & {3} & {0} \\\ {0} & {0} & {0} & {1}\end{array}\right]$$

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{array}{l}{3 x+2 y=0} \\ {-x-2 y=8}\end{array}\right.$$

Use back-substitution to solve the triangular system. $$ \left\\{\begin{aligned} 4 x+3 z &=10 \\ 2 y-z &=-6 \\ \frac{1}{2} z &=4 \end{aligned}\right. $$

Use the elimination method to find all solutions of the system of equations. \(\left\\{\begin{array}{l}{4 x-3 y=11} \\ {8 x+4 y=12}\end{array}\right.\)

1–14 Graph the inequality. $$y>x^{2}+1$$

Find the partial fraction decomposition of the rational function. \(\frac{2}{(x-1)(x+1)}\)

Find the inverse of the matrix if it exists. \(\left[\begin{array}{rr}{6} & {-3} \\ {-8} & {4}\end{array}\right]\)

Solve the matrix equation for the unknown matrix \(X\) , or explain why no solution exists. $$\begin{array}{l}{A=\left[\begin{array}{ll}{4} & {6} \\ {1} & {3}\end{array}\right] \quad B=\left[\begin{array}{ll}{2} & {5} \\ {3} & {7}\end{array}\right]} \\ {C=\left[\begin{array}{ll}{2} & {3} \\ {1} & {0} \\\ {0} & {2}\end{array}\right]} & {D=\left[\begin{array}{rr}{10} & {20} \\\ {10} & {0}\end{array}\right]}\end{array}$$ $$ 2 X+A=B $$

7–14 A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$\left[\begin{array}{llll}{1} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\\ {0} & {1} & {5} & {1}\end{array}\right]$$

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{aligned} x+3 y=5 \\ 2 x-y=3\end{aligned}\right.$$

Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. $$ \left\\{\begin{aligned} x-2 y-z &=4 \\ x-y+3 z &=0 \\ 2 x+y+z &=0 \end{aligned}\right. $$ Eliminate the \(x\) -term from the second equation.

Use the elimination method to find all solutions of the system of equations. \(\left\\{\begin{array}{l}{x^{2}-2 y=1} \\ {x^{2}+5 y=29}\end{array}\right.\)

1–14 Graph the inequality. $$x^{2}+y^{2} \geq 9$$

Find the partial fraction decomposition of the rational function. \(\frac{2 x}{(x-1)(x+1)}\)

\(9-14\) Evaluate the minor and cofactor using the matrix \(A\) $$ A=\left[\begin{array}{rrr}{1} & {0} & {\frac{1}{2}} \\ {-3} & {5} & {2} \\\ {0} & {0} & {4}\end{array}\right] $$ $$ M_{13}, A_{13} $$

Find the inverse of the matrix if it exists. \(\left[\begin{array}{ll}{\frac{1}{2}} & {\frac{1}{3}} \\ {5} & {4}\end{array}\right]\)

7–14 A matrix is given. (a) Determine whether the matrix is in row-echelon form. (b) Determine whether the matrix is in reduced row-echelon form. (c) Write the system of equations for which the given matrix is the augmented matrix. $$\left[\begin{array}{llll}{1} & {0} & {0} & {1} \\ {0} & {1} & {0} & {2} \\\ {0} & {0} & {1} & {3}\end{array}\right]$$

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered-pair form given in Example 3. $$\left\\{\begin{aligned} x+y &=7 \\ 2 x-3 y &=-1 \end{aligned}\right.$$

Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. $$ \left\\{\begin{aligned} x+y-3 z &=3 \\\\-2 x+3 y+z &=2 \\ x-y+2 z &=0 \end{aligned}\right. $$ Eliminate the \(x\) -term from the second equation

Use the elimination method to find all solutions of the system of equations. \(\left\\{\begin{array}{l}{3 x^{2}+4 y=17} \\ {2 x^{2}+5 y=2}\end{array}\right.\)