Chapter 4

Use an augmented matrix to solve each system. $$ \left\\{\begin{array}{l}{-x+5 y=15} \\ {2 x+3 y=9}\end{array}\right. $$

Shopping Suppose you want to fill nine 1-lb tins with a holiday snack mix. You plan to buy almonds for \(\$ 2.45 / 1 \mathrm{b}\) , peanuts for \(\$ 1.85 / 1 \mathrm{b}\) , and raisins for \(\$ .80 / 1 \mathrm{b}\) .You have \(\$ 15\) and want the mix to contain twice as much of the nuts as of the raisins by weight. How much of each ingredient should you buy? a. Writing Explain how each equation in the system at the right relates to the problem. What does each variable represent? b. Solve the system. $$ \left\\{\begin{array}{l}{x+y+z=9} \\ {2.45 x+1.85 y+0.8 z=15} \\ {x+y=2 z}\end{array}\right. $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rrr}{0} & {2} & {-3} \\ {1} & {2} & {4} \\ {-2} & {0} & {1}\end{array}\right] $$

Find the coordinates of each image after reflection in the given line. \(\left[\begin{array}{rrrrr}{-1} & {-2} & {-4} & {-6} & {-2} \\ {-4} & {0} & {0} & {-3} & {-4}\end{array}\right] ; x\) -axis

Find each product. $$ \left[\begin{array}{rr}{-3} & {5}\end{array}\right]\left[\begin{array}{rr}{0} & {-3} \\ {0} & {5}\end{array}\right] $$

Find the value of each variable. $$ \left[\begin{array}{rr}{2} & {2} \\ {-1} & {6}\end{array}\right]-\left[\begin{array}{rr}{4} & {-1} \\ {0} & {5}\end{array}\right]=\left[\begin{array}{rr}{x} & {y} \\ {-1} & {z}\end{array}\right] $$

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{ll}{1} & {3} \\ {2} & {0}\end{array}\right] $$

Use an augmented matrix to solve each system. $$ \left\\{\begin{array}{l}{3 x+6 y=2} \\ {2 x-y=3}\end{array}\right. $$

Determine whether each system has a unique solution.$$ \left\\{\begin{array}{l}{20 x+5 y=240} \\ {y=20 x}\end{array}\right. $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rrr}{5} & {1} & {0} \\ {0} & {2} & {-1} \\ {-2} & {-3} & {1}\end{array}\right] $$

Find each product. $$ \left[\begin{array}{rr}{0} & {-3} \\ {0} & {5}\end{array}\right]\left[\begin{array}{rr}{-3} & {0} \\ {5} & {0}\end{array}\right] $$

Find the value of each variable. $$ \left[\begin{array}{cc}{2} & {4} \\ {8} & {12}\end{array}\right]=\left[\begin{array}{cc}{4 x-6} & {-10 t+5 x} \\ {4 x} & {15 t+1.5 x}\end{array}\right] $$

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{rr}{6} & {-8} \\ {-3} & {4}\end{array}\right] $$

Solve each system. $$ \left\\{\begin{aligned} x+y+z &=2 \\ 2 y-2 z &=2 \\ x &-3 z=1 \end{aligned}\right. $$

Determine whether each system has a unique solution.$$ \left\\{\begin{array}{l}{20 x+5 y=145} \\ {30 x-5 y=125}\end{array}\right. $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rrr}{4} & {6} & {-1} \\ {2} & {3} & {2} \\ {1} & {-1} & {1}\end{array}\right] $$

Find each product. $$ \left[\begin{array}{rr}{1} & {0} \\ {-1} & {-5} \\ {0} & {3}\end{array}\right]\left[\begin{array}{rr}{-1} & {0} \\ {0} & {-1}\end{array}\right] $$

Find each matrix sum or difference if possible. If not possible, explain. $$A=\left[\begin{array}{rr}{3} & {4} \\ {6} & {-2} \\ {1} & {0}\end{array}\right] \quad B=\left[\begin{array}{rr}{-3} & {1} \\ {2} & {-4} \\\ {-1} & {5}\end{array}\right] \quad C=\left[\begin{array}{rr}{1} & {2} \\\ {-3} & {1}\end{array}\right] \quad D=\left[\begin{array}{cc}{5} & {1} \\ {0} & {2}\end{array}\right]$$ \(A+B\)

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{rr}{4} & {8} \\ {-3} & {-2}\end{array}\right] $$

Solve each system. $$ \left\\{\begin{aligned} x-y+z &=3 \\ x+3 z &=6 \\ y-2 z &=-1 \end{aligned}\right. $$

Determine whether each system has a unique solution. $$ \left\\{\begin{array}{l}{y=2000-65 x} \\ {y=500+55 x}\end{array}\right. $$

Evaluate the determinant of each matrix. $$ \left[\begin{array}{rrr}{-3} & {2} & {-1} \\ {2} & {5} & {2} \\ {1} & {-2} & {0}\end{array}\right] $$

Business A florist makes three special floral arrangements. One uses three lilies. The second uses three lilies and four carnations. The third uses four daisies and three carnations. Lilies cost \(\$ 2.15\) each, carnations cost \(\$ .90\) each, and daisies cost \(\$ 1.30\) each. a. Write a matrix to show the number of each type of flower in each arrangement. b. Write a matrix to show the cost of each type of flower. c. Find the matrix showing the cost of each floral arrangement.

Find each matrix sum or difference if possible. If not possible, explain. $$A=\left[\begin{array}{rr}{3} & {4} \\ {6} & {-2} \\ {1} & {0}\end{array}\right] \quad B=\left[\begin{array}{rr}{-3} & {1} \\ {2} & {-4} \\\ {-1} & {5}\end{array}\right] \quad C=\left[\begin{array}{rr}{1} & {2} \\\ {-3} & {1}\end{array}\right] \quad D=\left[\begin{array}{cc}{5} & {1} \\ {0} & {2}\end{array}\right]$$ \(B+D\)

Describe the information necessary to make a matrix containing numerical data meaningful.

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{rr}{-1.5} & {3} \\ {2.5} & {-0.5}\end{array}\right] $$

Solve each system. $$ \left\\{\begin{aligned} x+y-z &=-1 \\ 3 x+4 y-z &=8 \\ 6 x+8 y-2 z &=16 \end{aligned}\right. $$

Determine whether each system has a unique solution. $$ \left\\{\begin{array}{l}{y=\frac{2}{3} x-3} \\ {y=-x+7}\end{array}\right. $$

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{rrr}{-2} & {1} & {-1} \\ {2} & {0} & {4} \\ {0} & {2} & {5}\end{array}\right] $$

Determine whether each product is defined or undefined. $$F=\left[\begin{array}{ll}{2} & {3} \\ {6} & {9}\end{array}\right] \quad G=\left[\begin{array}{rr}{-3} & {6} \\ {2} & {-4}\end{array}\right] \quad H=\left[\begin{array}{r}{-5} \\ {6}\end{array}\right] \quad J=\left[\begin{array}{ll}{0} & {7}\end{array}\right]$$ $$ F G $$

Find each matrix sum or difference if possible. If not possible, explain. $$A=\left[\begin{array}{rr}{3} & {4} \\ {6} & {-2} \\ {1} & {0}\end{array}\right] \quad B=\left[\begin{array}{rr}{-3} & {1} \\ {2} & {-4} \\\ {-1} & {5}\end{array}\right] \quad C=\left[\begin{array}{rr}{1} & {2} \\\ {-3} & {1}\end{array}\right] \quad D=\left[\begin{array}{cc}{5} & {1} \\ {0} & {2}\end{array}\right]$$ \(C+D\)

State the dimensions of each matrix. Identify the indicated element. \(\left[\begin{array}{rrr}{4} & {6} & {5} \\ {2} & {-3} & {-7} \\ {1} & {0} & {9}\end{array}\right], a_{23}\)

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{rr}{1} & {-2} \\ {3} & {0}\end{array}\right] $$

Solve each system. $$ \left\\{\begin{aligned} x+y-z &=1 \\ 3 x+3 y+z &=3 \\ 2 x+2 y-2 z &=2 \end{aligned}\right. $$

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{rrr}{2} & {0} & {-1} \\ {-1} & {-1} & {1} \\ {3} & {2} & {0}\end{array}\right] $$

Find the coordinates of each image after the given rotation. $$ \left[\begin{array}{llll}{0} & {4} & {8} & {6} \\ {0} & {4} & {4} & {2}\end{array}\right] ; 360^{\circ} $$

Determine whether each product is defined or undefined. $$F=\left[\begin{array}{ll}{2} & {3} \\ {6} & {9}\end{array}\right] \quad G=\left[\begin{array}{rr}{-3} & {6} \\ {2} & {-4}\end{array}\right] \quad H=\left[\begin{array}{r}{-5} \\ {6}\end{array}\right] \quad J=\left[\begin{array}{ll}{0} & {7}\end{array}\right]$$ $$ G F $$

Find each matrix sum or difference if possible. If not possible, explain. $$A=\left[\begin{array}{rr}{3} & {4} \\ {6} & {-2} \\ {1} & {0}\end{array}\right] \quad B=\left[\begin{array}{rr}{-3} & {1} \\ {2} & {-4} \\\ {-1} & {5}\end{array}\right] \quad C=\left[\begin{array}{rr}{1} & {2} \\\ {-3} & {1}\end{array}\right] \quad D=\left[\begin{array}{cc}{5} & {1} \\ {0} & {2}\end{array}\right]$$ \(B-A\)

State the dimensions of each matrix. Identify the indicated element. \(\left[\begin{array}{rrr}{-4} & {1} & {-3} \\ {2} & {1} & {0}\end{array}\right], a_{12}\)

Solve each matrix equation. If an equation cannot be solved, explain why. $$ \left[\begin{array}{cc}{12} & {7} \\ {5} & {3}\end{array}\right] X=\left[\begin{array}{rr}{2} & {-1} \\ {3} & {2}\end{array}\right] $$

Solve each system. $$ \left\\{\begin{aligned} x+y &=1 \\ y+z &=2 \\ x &-z=-1 \end{aligned}\right. $$

Determine whether each system has a unique solution. $$ \left\\{\begin{array}{l}{x+2 y+z=4} \\ {y=x-3} \\ {z=2 x}\end{array}\right. $$

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{rrr}{0} & {0} & {2} \\ {1} & {4} & {-2} \\ {3} & {-2} & {1}\end{array}\right] $$

Find the coordinates of each image after the given rotation. $$ \left[\begin{array}{lllll}{1} & {2} & {3} & {4} & {2.5} \\ {3} & {2} & {2} & {3} & {5}\end{array}\right] ; 180^{\circ} $$

Determine whether each product is defined or undefined. $$F=\left[\begin{array}{ll}{2} & {3} \\ {6} & {9}\end{array}\right] \quad G=\left[\begin{array}{rr}{-3} & {6} \\ {2} & {-4}\end{array}\right] \quad H=\left[\begin{array}{r}{-5} \\ {6}\end{array}\right] \quad J=\left[\begin{array}{ll}{0} & {7}\end{array}\right]$$ $$ F H $$

Find each matrix sum or difference if possible. If not possible, explain. $$A=\left[\begin{array}{rr}{3} & {4} \\ {6} & {-2} \\ {1} & {0}\end{array}\right] \quad B=\left[\begin{array}{rr}{-3} & {1} \\ {2} & {-4} \\\ {-1} & {5}\end{array}\right] \quad C=\left[\begin{array}{rr}{1} & {2} \\\ {-3} & {1}\end{array}\right] \quad D=\left[\begin{array}{cc}{5} & {1} \\ {0} & {2}\end{array}\right]$$ \(C-D\)

State the dimensions of each matrix. Identify the indicated element. \(\left[\begin{array}{lll}{1} & {1} & {1} \\ {1} & {0} & {0} \\ {1} & {0} & {0}\end{array}\right], a_{32}\)

Solve each matrix equation. If an equation cannot be solved, explain why. $$ \left[\begin{array}{cc}{0} & {-4} \\ {0} & {-1}\end{array}\right] X=\left[\begin{array}{l}{0} \\ {4}\end{array}\right] $$

Solve each system. $$ \left\\{\begin{aligned} x &+z=-4 \\ y-z &=1 \\ x+y &=-3 \end{aligned}\right. $$

Solve each matrix equation. If the coefficient matrix has no inverse, write \(n o\) unique solution. $$ \left[\begin{array}{ll}{1} & {1} \\ {1} & {2}\end{array}\right]\left[\begin{array}{l}{x} \\\ {y}\end{array}\right]=\left[\begin{array}{c}{8} \\ {10}\end{array}\right] $$