Chapter 10

1–14 Graph the inequality. $$x<3$$

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

\(1-8\) Find the determinant of the matrix, if it exists. $$ \left[\begin{array}{ll}{2} & {0} \\ {0} & {3}\end{array}\right] $$

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A .\) \(A=\left[\begin{array}{ll}{4} & {1} \\ {7} & {2}\end{array}\right], \quad B=\left[\begin{array}{rr}{2} & {-1} \\ {-7} & {4}\end{array}\right]\)

Determine whether the matrices \(A\) and \(B\) are equal. $$ A=\left[\begin{array}{rrr}{1} & {-2} & {0} \\ {\frac{1}{2}} & {6} & {0}\end{array}\right], \quad B=\left[\begin{array}{rr}{1} & {-2} \\\ {\frac{1}{2}} & {6}\end{array}\right] $$

1–6 State the dimension of the matrix. $$\left[\begin{array}{cc}{2} & {7} \\ {0} & {-1} \\ {5} & {-3}\end{array}\right]$$

State whether the equation or system of equations is linear. $$ 6 x-\sqrt{3} y+\frac{1}{2} z=0 $$

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{array}{c}{x+y=4} \\ {2 x-y=2}\end{array}\right.$$

Use the substitution method to find all solutions of the system of equations. \(\left\\{\begin{aligned} x-y &=2 \\ 2 x+3 y &=9 \end{aligned}\right.\)

1–14 Graph the inequality. $$y \geq-2$$

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

\(1-8\) Find the determinant of the matrix, if it exists. $$ \left[\begin{array}{rr}{0} & {-1} \\ {2} & {0}\end{array}\right] $$

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A .\) \(A=\left[\begin{array}{ll}{2} & {-3} \\ {4} & {-7}\end{array}\right], \quad B=\left[\begin{array}{ll}{\frac{7}{2}} & {-\frac{3}{2}} \\ {2} & {-1}\end{array}\right]\)

Determine whether the matrices \(A\) and \(B\) are equal. $$ A=\left[\begin{array}{cc}{\frac{1}{4}} & {\ln 1} \\ {2} & {3}\end{array}\right], \quad B=\left[\begin{array}{cc}{0.25} & {0} \\\ {\sqrt{4}} & {\frac{6}{2}}\end{array}\right] $$

1–6 State the dimension of the matrix. $$\left[\begin{array}{rrrr}{-1} & {5} & {4} & {0} \\ {0} & {2} & {11} & {3}\end{array}\right]$$

State whether the equation or system of equations is linear. $$ x^{2}+y^{2}+z^{2}=4 $$

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{array}{c}{2 x+y=11} \\ {x-2 y=4}\end{array}\right.$$

Use the substitution method to find all solutions of the system of equations. \(\left\\{\begin{array}{c}{2 x+y=7} \\ {x+2 y=2}\end{array}\right.\)

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

\(1-8\) Find the determinant of the matrix, if it exists. $$ \left[\begin{array}{rr}{4} & {5} \\ {0} & {-1}\end{array}\right] $$

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A .\) \(A=\left[\begin{array}{rrr}{1} & {3} & {-1} \\ {1} & {4} & {0} \\ {-1} & {-3} & {2}\end{array}\right], \quad B=\left[\begin{array}{rrr}{8} & {-3} & {4} \\\ {-2} & {1} & {-1} \\ {1} & {0} & {1}\end{array}\right]\)

Perform the matrix operation, or if it is impossible, explain why. $$ \left[\begin{array}{rr}{2} & {6} \\ {-5} & {3}\end{array}\right]+\left[\begin{array}{rr}{-1} & {-3} \\ {6} & {2}\end{array}\right] $$

1–6 State the dimension of the matrix. $$\left[\begin{array}{l}{12} \\ {35}\end{array}\right]$$

State whether the equation or system of equations is linear. $$ \left\\{\begin{aligned} x y-3 y+z &=5 \\ x-y^{2}+5 z &=0 \\ 2 x+y z &=3 \end{aligned}\right. $$

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{array}{l}{2 x-3 y=12} \\ {-x+\frac{3}{2} y=4}\end{array}\right.$$

Use the substitution method to find all solutions of the system of equations. \(\left\\{\begin{array}{l}{y=x^{2}} \\ {y=x+12}\end{array}\right.\)

1–14 Graph the inequality. $$y < x+2$$

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

\(1-8\) Find the determinant of the matrix, if it exists. $$ \left[\begin{array}{rr}{-2} & {1} \\ {3} & {-2}\end{array}\right] $$

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A .\) \(A=\left[\begin{array}{rrr}{3} & {2} & {4} \\ {1} & {1} & {-6} \\ {2} & {1} & {12}\end{array}\right], \quad B=\left[\begin{array}{rrr}{9} & {-10} & {-8} \\\ {-12} & {14} & {11} \\ {-\frac{1}{2}} & {\frac{1}{2}} & {\frac{1}{2}}\end{array}\right]\)

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

1–6 State the dimension of the matrix. $$\left[\begin{array}{r}{-3} \\ {0} \\ {1}\end{array}\right]$$

State whether the equation or system of equations is linear. $$ \left\\{\begin{aligned} x-2 y+3 z &=10 \\ 2 x+5 y &=2 \\ y+2 z &=4 \end{aligned}\right. $$

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{aligned} 2 x+6 y &=0 \\\\-3 x-9 y &=18 \end{aligned}\right.$$

Use the substitution method to find all solutions of the system of equations. \(\left\\{\begin{aligned} x^{2}+y^{2} &=25 \\ y &=2 x \end{aligned}\right.\)

1–14 Graph the inequality. $$y \leq 2 x+2$$

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

Find the inverse of the matrix and verify that \(A^{-1} A=A A^{-1}=I_{2}\) and \(B^{-1} B=B B^{-1}=I_{3} .\) \(A=\left[\begin{array}{ll}{7} & {4} \\ {3} & {2}\end{array}\right]\)

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

1–6 State the dimension of the matrix. $$\left[\begin{array}{lll}{1} & {4} & {7}\end{array}\right]$$

$$ \left\\{\begin{aligned} x-2 y+4 z &=3 \\ y+2 z &=7 \\ z &=2 \end{aligned}\right. $$

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{array}{l}{-x+\frac{1}{2} y=-5} \\ {2 x-y=10}\end{array}\right.$$

Use the substitution method to find all solutions of the system of equations. \(\left\\{\begin{array}{l}{x^{2}+y^{2}=8} \\ {x+y=0}\end{array}\right.\)

1–14 Graph the inequality. $$y<-x+5$$

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

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

1–6 State the dimension of the matrix. $$\left[\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right]$$

Use back-substitution to solve the triangular system. $$ \left\\{\begin{aligned} x+y-3 z &=8 \\ y-3 z &=5 \\ z &=-1 \end{aligned}\right. $$

Graph each linear system, either by hand or using a graphing device. Use the graph to determine if the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{aligned} 12 x+15 y &=-18 \\ 2 x+\frac{5}{2} y &=-3 \end{aligned}\right.$$

Use the substitution method to find all solutions of the system of equations. \(\left\\{\begin{aligned} x^{2}+y &=9 \\ x-y+3 &=0 \end{aligned}\right.\)