Chapter 10

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. $$\left[\begin{array}{rrr} 1 & 3 & -3 \\ 0 & 1 & 5 \end{array}\right]$$

Use the elimination method to find all solutions of the system of equations. $$\left\\{\begin{array}{c} x^{2}-y^{2}=1 \\ 2 x^{2}-y^{2}=x+3 \end{array}\right.$$

Finding the Inverse of a Matrix Find the inverse of the matrix if it exists. $$\left[\begin{array}{rr}-7 & 4 \\\8 & -5\end{array}\right]$$

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

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

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system. (Graph cannot copy) $$\left\\{\begin{aligned} x+y &=2 \\ 2 x+y &=5 \end{aligned}\right.$$

Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. \(\left\\{\begin{array}{rr}-5 x+2 y-3 z= & 3 \\ 10 x-3 y+z= & -20 \\ -x+3 y+z= & 8\end{array}\right.\) Eliminate the \(x\)-term from the second equation.

Find the partial fraction decomposition of the rational function. $$\frac{5}{(x-1)(x+4)}$$

Graph the inequality. $$y>x-3$$

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. $$\left[\begin{array}{llll} 1 & 2 & 8 & 0 \\ 0 & 1 & 3 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system. $$\left\\{\begin{array}{l} x^{2}+y=8 \\ x-2 y=-6 \end{array}\right.$$ (GRAPH CANNOT COPY)

Finding the Inverse of a Matrix Find the inverse of the matrix if it exists. $$\left[\begin{array}{rr}6 & -3 \\\\-8 & 4\end{array}\right]$$

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

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

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

Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. \(\left\\{\begin{array}{l}2 x+y-3 z=5 \\ 2 x+3 y+z=13 \\ 6 x-5 y-z=7\end{array}\right.\) Eliminate the \(x\)-term from the third equation.

Find the partial fraction decomposition of the rational function. $$\frac{x+6}{x(x+3)}$$

Graph the inequality. $$y \leq 1-x$$

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. $$\left[\begin{array}{rrrr} 1 & 0 & -7 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system. $$\left\\{\begin{array}{l} x-y^{2}=-4 \\ x-y=2 \end{array}\right.$$ (GRAPH CANNOT COPY)

Finding the Inverse of a Matrix Find the inverse of the matrix if it exists. $$\left[\begin{array}{ll}\frac{1}{2} & \frac{1}{3} \\\5 & 4\end{array}\right]$$

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

Graph each linear system, either by hand or using a graphing device. Use the graph to determine whether the system has one solution, no solution, or infinitely many solutions. If there is exactly one solution, use the graph to find it. $$\left\\{\begin{array}{l} 2 x-y=4 \\ 3 x+y=6 \end{array}\right.$$

Perform an operation on the given system that eliminates the indicated variable. Write the new equivalent system. \(\left\\{\begin{aligned} x-3 y+2 z &=-1 \\ y+z &=-1 \\ 2 y-z &=1 \end{aligned}\right.\) Eliminate the \(y\) -term from the third equation.

Find the partial fraction decomposition of the rational function. $$\frac{12}{x^{2}-9}$$

Graph the inequality. $$2 x-y \geq-4$$

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. $$\left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 5 & 1 \end{array}\right]$$

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system. $$\left\\{\begin{aligned} x^{2}+y &=0 \\ x^{3}-2 x-y &=0 \end{aligned}\right.$$ (GRAPH CANNOT COPY)

Finding the Inverse of a Matrix Find the inverse of the matrix if it exists. $$\left[\begin{array}{rr}0.4 & -1.2 \\\0.3 & 0.6\end{array}\right]$$

Evaluate the minor and cofactor using the matrix \(A\). $$A=\left[\begin{array}{rrr} 1 & 0 & \frac{1}{2} \\ -3 & 5 & 2 \\ 0 & 0 & 4 \end{array}\right]$$ $$M_{12}, A_{12}$$

Solve the matrix equation for the unknown matrix \(X\), or explain why no solution exists. $$\begin{array}{c}A=\left[\begin{array}{ll}4 & 6 \\\1 & 3\end{array}\right] & B=\left[\begin{array}{ll}2 & 5 \\\3 & 7\end{array}\right] \\\C=\left[\begin{array}{ll}2 & 3 \\\1 & 0 \\\0 & 2\end{array}\right] & D=\left[\begin{array}{lr}10 & 20 \\\30 & 20 \\\10 & 0\end{array}\right]\end{array}$$ $$2 X+A=B$$

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

Find the complete solution of the linear system, or show that it is inconsistent. \(\left\\{\begin{array}{rr}x-y-z= & 4 \\ 2 y+z= & -1 \\ -x+y-2 z= & 5\end{array}\right.\)

Find the partial fraction decomposition of the rational function. $$\frac{x-12}{x^{2}-4 x}$$

Graph the inequality. $$3 x-y-9<0$$

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. $$\left[\begin{array}{llll} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \end{array}\right]$$

Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system. $$\left\\{\begin{aligned} x^{2}+y^{2} &=4 x \\ x &=y^{2} \end{aligned}\right.$$ (GRAPH CANNOT COPY

Finding the Inverse of a Matrix Find the inverse of the matrix if it exists. $$\left[\begin{array}{lll}4 & 2 & 3 \\\3 & 3 & 2 \\\1 & 0 & 1\end{array}\right]$$

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

Solve the matrix equation for the unknown matrix \(X\), or explain why no solution exists. $$\begin{array}{c}A=\left[\begin{array}{ll}4 & 6 \\\1 & 3\end{array}\right] & B=\left[\begin{array}{ll}2 & 5 \\\3 & 7\end{array}\right] \\\C=\left[\begin{array}{ll}2 & 3 \\\1 & 0 \\\0 & 2\end{array}\right] & D=\left[\begin{array}{lr}10 & 20 \\\30 & 20 \\\10 & 0\end{array}\right]\end{array}$$ $$3 X-B=C$$

$$\left\\{\begin{aligned} 2 x+6 y &=0 \\ -3 x-9 y &=18 \end{aligned}\right.$$

Find the complete solution of the linear system, or show that it is inconsistent. \(\left\\{\begin{aligned} x-y+z &=0 \\ y+2 z &=-2 \\ x+y-z &=2 \end{aligned}\right.\)

Find the partial fraction decomposition of the rational function. $$\frac{4}{x^{2}-4}$$

Graph the inequality. $$-x^{2}+y \geq 5$$

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. $$\left[\begin{array}{rrrrr} 1 & 3 & 0 & -1 & 0 \\ 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$$

Find all solutions of the system of equations. $$\left\\{\begin{array}{l} y+x^{2}=4 x \\ y+4 x=16 \end{array}\right.$$

Finding the Inverse of a Matrix Find the inverse of the matrix if it exists. $$\left[\begin{array}{rrr}2 & 4 & 1 \\\\-1 & 1 & -1 \\\1 & 4 & 0\end{array}\right]$$

Evaluate the minor and cofactor using the matrix \(A\). $$A=\left[\begin{array}{rrr} 1 & 0 & \frac{1}{2} \\ -3 & 5 & 2 \\ 0 & 0 & 4 \end{array}\right]$$ $$M_{23}, A_{23}$$

Solve the matrix equation for the unknown matrix \(X\), or explain why no solution exists. $$\begin{array}{c}A=\left[\begin{array}{ll}4 & 6 \\\1 & 3\end{array}\right] & B=\left[\begin{array}{ll}2 & 5 \\\3 & 7\end{array}\right] \\\C=\left[\begin{array}{ll}2 & 3 \\\1 & 0 \\\0 & 2\end{array}\right] & D=\left[\begin{array}{lr}10 & 20 \\\30 & 20 \\\10 & 0\end{array}\right]\end{array}$$ $$2(B-X)=D$$

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