Chapter 4

A manufacturer sells pencil and erasers in packages. The price of a package of five erasers and two pencils is \(\$ .23 .\) The price of a package of seven erasers and five pencils is \(\$ .41 .\) Find the price of one eraser and one pencil.

Solve each system. $$ \left\\{\begin{aligned} x+y+z &=4 \\ 4 x+5 y &=4 \\ y-3 z &=-9 \end{aligned}\right. $$

The matrix \(A\) at the right has an inverse \(A^{-1}\) . What is the product \(A A^{-1} ?\) Explain. $$ A=\left[\begin{array}{rrr}{1} & {1} & {2} \\ {2} & {-1} & {1} \\ {1} & {4} & {-1}\end{array}\right] $$

Mental Math Find each product. $$ 0.5\left[\begin{array}{cc}{3} & {14} \\ {7} & {-4}\end{array}\right] $$

Anna made a table to show how much money she and two of her friends earned for summer chores. $$\begin{array}{|c|c|c|c|}\hline \text { Anna } & {\$ 20} & {\$ 40} & {\$ 0} \\\ \hline \text { Rob } & {\$ 12} & {\$ 35} & {\$ 40} \\ \hline \text { Carla } & {\$ 15} & {\$ 55} & {\$ 70} \\ \hline\end{array}$$ a. Display the data in Anna's table in a matrix A with each row representing someone's earnings. b. Rob made his own matrix, \(R,\) to show the earnings. \(R=\left[\begin{array}{rrr}{12} & {20} & {15} \\ {35} & {40} & {55} \\ {40} & {0} & {70}\end{array}\right]\). What do values in the first column of matrix \(R\) represent?

Find the sum of \(E=\left[\begin{array}{l}{3} \\ {4} \\\ {7}\end{array}\right]\) and the additive inverse of \(G=\left[\begin{array}{r}{-2} \\ {0} \\ {5}\end{array}\right]\)

Multiple choice Suppose you invested \(\$ 5000\) in three different funds for one year. The funds paid simple interest of \(8 \%, 10 \%\) , and 7\(\%\) , respectively. The total interest at the end of one year was \(\$ 405 .\) You invested \(\$ 500\) more at 10\(\%\) than at 8\(\%\) . How much did you invest in the 10\(\%\) fund? \(\begin{array}{ll}{\text { A } \$ 150} & {\text { B } \$ 1000}\end{array}\) C \(\$ 1500 \quad\) D \(\$ 2500\)

Determine whether the matrices are multiplicative inverses. If they are not, explain why not. $$ \left[\begin{array}{cc}{-2} & {-5} \\ {-2} & {-4}\end{array}\right],\left[\begin{array}{cc}{-2.5} & {2} \\ {1} & {-1}\end{array}\right] $$

Solve each system. $$ \left\\{\begin{aligned} x+y+z &=4 \\ 4 x+5 y &=3 \\ y-3 z &=-10 \end{aligned}\right. $$

Matrices \(B\) and \(C\) are inverses of each other. $$ B=\left[\begin{array}{ccc}{1} & {1} & {0} \\ {0} & {2} & {1} \\ {-2} & {0} & {2}\end{array}\right] \quad C=\left[\begin{array}{rrr}{2} & {-1} & {0.5} \\\ {-1} & {1} & {-0.5} \\ {2} & {-1} & {1}\end{array}\right] \quad D=\left[\begin{array}{c}{2} \\ {0} \\ {-1}\end{array}\right] $$ Solve the matrix equation \(B X=D\)

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

Solve each system. \(\left\\{\begin{aligned} 3 x+2 y-2 z &=-9 \\ 5 x &-3 z=&-7 \\ x+4 y+3 z &=5 \end{aligned}\right.\)

Prove that matrix addition is commutative for \(2 \times 2\) matrices.

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{rr}{-9} & {3} \\ {4} & {2.5}\end{array}\right] $$

Solve each system. $$ \left\\{\begin{aligned}-2 w+x+y &=0 \\\\-w+2 x-y+z &=1 \\\\-2 w+3 x+3 y+2 z &=6 \\ w+x+2 y+z &=5 \end{aligned}\right. $$

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

Solve each system. \(\left\\{\begin{aligned} 2 x+3 y+4 z &=-1 \\ x-2 y+z &=9 \\ x+4 y-2 z &=-12 \end{aligned}\right.\)

Prove that matrix addition is associative for \(2 \times 2\) matrices.

Solve each system. $$\left\\{\begin{aligned} 2 x-3 y+2 z &=10 \\ x+3 y+4 z &=14 \\ 3 x-y+z &=9 \end{aligned}\right.$$

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{cc}{2} & {3} \\ {8} & {12}\end{array}\right] $$

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

What is true about $$\left[\begin{array}{rr}{-3} & {0} \\ {0} & {-3}\end{array}\right]\left[\begin{array}{rrr}{1} & {-2} & {4} \\ {1} & {-1} & {2}\end{array}\right] and -3\left[\begin{array}{rrr}{1} & {-2} & {4} \\ {1} & {-1} & {2}\end{array}\right]$$ $$\begin{array}{l}{\text { A They are equal matrices. }} \\ {\text { C Both are matrices for reflection. }}\end{array}$$ $$ \begin{array}{l}{\text { B They are opposite matrices. }} \\ {\text { D Neither product exists. }}\end{array} $$

Multiple choice Columns in matrix \(A=\left[\begin{array}{cc}{3} & {8} \\ {0} & {12}\end{array}\right]\) show, respectively, the numbers of erasers and pencils sold. The rows how, respectively, the numbers sold on Monday and Tuesday. Matrix \(B=\left[\begin{array}{c}{0.50} \\\ {0.25}\end{array}\right]\) shows the 50 cost of one eraser and the quarter cost of one pencil. What does the product \(A B\) show? (A) the total paid for erasers on Monday and Tuesday and the total paid for pencils on Monday and Tuesday (B) the total paid for erasers and pencils on Monday and the total paid for erasers and pencils on Tuesday. (C) the total paid for pencils and erasers (D) the cost of 1 pencil and 1 eraser

Solve each system by graphing. \(\left\\{\begin{aligned} 2 x+y &=8 \\ x-3 y &=-3 \end{aligned}\right.\)

Use matrices \(A=\left[\begin{array}{rrr}{5} & {7} & {3} \\ {-1} & {0} & {-4}\end{array}\right]\) and \(C=\left[\begin{array}{rrr}{-7} & {4} & {2} \\\ {1} & {-2} & {-3}\end{array}\right]\) for Exercises 35 and 36 What is the sum \(A+C ?\) A. The matrices cannot be added. \(\mathbf{B} \cdot\left[\begin{array}{rrr}{-2} & {11} & {5} \\ {0} & {-2} & {-7}\end{array}\right]\) \(C .\left[\begin{array}{ccc}{12} & {3} & {1} \\ {-2} & {2} & {-1}\end{array}\right]\) \(D \cdot\left[\begin{array}{ccc}{-35} & {28} & {6} \\ {-1} & {0} & {12}\end{array}\right]\)

Solve each system. $$ \left\\{\begin{aligned} 4 x-y+z &=3 \\ x+2 y+z &=0 \\ 3 x+7 y-3 z &=6 \end{aligned}\right. $$

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{rr}{-3} & {4} \\ {9} & {10}\end{array}\right] $$

The coordinates \((x, y)\) of a point in a plane are the solution of the system \(\left\\{\begin{array}{l}{2 x+3 y=13} \\ {5 x+7 y=31}\end{array} . \text { Find the coordinates of the point. }\right.\)

Find the dimensions of each product matrix. Then find each product. $$ \left[\begin{array}{rrr}{5} & {7} & {0} \\ {-\frac{4}{5}} & {3} & {6} \\ {0} & {-\frac{2}{3}} & {4}\end{array}\right]\left[\begin{array}{rr}{2} & {-1} \\\ {1} & {1} \\ {0} & {-1}\end{array}\right] $$

Solve each system by graphing. \(\left\\{\begin{array}{c}{2 x+y=7} \\ {x+y=-5}\end{array}\right.\)

Use matrices \(A=\left[\begin{array}{rrr}{5} & {7} & {3} \\ {-1} & {0} & {-4}\end{array}\right]\) and \(C=\left[\begin{array}{rrr}{-7} & {4} & {2} \\\ {1} & {-2} & {-3}\end{array}\right]\) for Exercises 35 and 36 What is matrix \(Y\) if \(Y-A=\left[\begin{array}{lll}{1} & {0} & {1} \\ {0} & {1} & {0}\end{array}\right] ?\) \(F \cdot\left[\begin{array}{rrr}{4} & {7} & {2} \\ {-1} & {-1} & {-5}\end{array}\right]\) G. \(\left[\begin{array}{rrr}{6} & {7} & {4} \\ {-1} & {1} & {-4}\end{array}\right]\) \(\mathrm{H} \cdot\left[\begin{array}{rrr}{-6} & {4} & {3} \\ {1} & {-1} & {-3}\end{array}\right]\) J. \(\left[\begin{array}{rrr}{-4} & {-7} & {-2} \\ {1} & {1} & {5}\end{array}\right]\)

Solve each system. $$ \left\\{\begin{array}{l}{x+2 y+z=4} \\ {3 x+6 y+3 z=2} \\\ {x-y+z=3}\end{array}\right. $$

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

A rectangle is twice as long as it is wide. The perimeter is 840 \(\mathrm{ft}\) . Find the dimensions of the rectangle.

Solve each system by graphing. \(\left\\{\begin{array}{c}{x+6 y=7} \\ {2 x+4 y=-2}\end{array}\right.\)

Find the value of each variable. $$ \left[\begin{array}{cc}{x} & {y-2} \\ {z} & {w+4}\end{array}\right]+\left[\begin{array}{rr}{2} & {5} \\ {-2} & {4}\end{array}\right]=\left[\begin{array}{ll}{6} & {1} \\ {4} & {8}\end{array}\right] $$

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. If it does not exist, explain why not. $$ \left[\begin{array}{ll}{0} & {3} \\ {3} & {0}\end{array}\right] $$

Complete each system for the given number of solutions. $$ \begin{array}{l}{\text { infinitely many }} \\\ {\left\\{\begin{array}{c}{x+y=7} \\ {2 x+2 y=\square}\end{array}\right.}\end{array} $$

Solve each system of equations. $$ \left\\{\begin{array}{l}{2 x+2 y=10} \\ {3 x-y=4}\end{array}\right. $$

For Exercises \(38-45,\) use matrices \(D, E,\) and \(F\) shown below. Perform the indicated operations if they are defined. If an operation is not defined, label it undefined. $$D=\left[\begin{array}{rrr}{1} & {2} & {-1} \\ {0} & {3} & {1} \\ {2} & {-1} & {-2}\end{array}\right] \quad E=\left[\begin{array}{rrr}{2} & {-5} & {0} \\\ {1} & {0} & {-2} \\ {3} & {1} & {1}\end{array}\right] \quad F=\left[\begin{array}{rr}{-3} & {2} \\ {-5} & {1} \\ {2} & {4}\end{array}\right]$$ $$ D E $$

Find the constant of variation for a direct variation that includes the given values. \((2,4)\)

Find the value of each variable. $$ \left[\begin{array}{rr}{x} & {3} \\ {x} & {-2}\end{array}\right]+\left[\begin{array}{rr}{y} & {6} \\ {-y} & {3}\end{array}\right]=\left[\begin{array}{ll}{6} & {9} \\ {4} & {1}\end{array}\right] $$

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. If it does not exist, explain why not. $$ \left[\begin{array}{rr}{-1} & {3} \\ {2} & {0}\end{array}\right] $$

Solve each system of equations. $$ \left\\{\begin{array}{l}{-x+y+z=5} \\ {2 x+y-z=2} \\ {3 x+2 y+4 z=0}\end{array}\right. $$

For Exercises \(38-45,\) use matrices \(D, E,\) and \(F\) shown below. Perform the indicated operations if they are defined. If an operation is not defined, label it undefined. $$D=\left[\begin{array}{rrr}{1} & {2} & {-1} \\ {0} & {3} & {1} \\ {2} & {-1} & {-2}\end{array}\right] \quad E=\left[\begin{array}{rrr}{2} & {-5} & {0} \\\ {1} & {0} & {-2} \\ {3} & {1} & {1}\end{array}\right] \quad F=\left[\begin{array}{rr}{-3} & {2} \\ {-5} & {1} \\ {2} & {4}\end{array}\right]$$ $$ -3 F $$

Find the constant of variation for a direct variation that includes the given values. \((-1,7)\)

Solve using Cramer's Rule. (Hint: Start by substituting \(m=\frac{1}{x}\) and \(n=\frac{1}{y}\) .) $$ \left\\{\begin{array}{l}{\frac{4}{x}+\frac{1}{y}=1} \\\ {\frac{8}{x}+\frac{4}{y}=3}\end{array}\right. $$

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. If it does not exist, explain why not. $$ \left[\begin{array}{ll}{1} & {2} \\ {2} & {1}\end{array}\right] $$

For Exercises \(38-45,\) use matrices \(D, E,\) and \(F\) shown below. Perform the indicated operations if they are defined. If an operation is not defined, label it undefined. $$D=\left[\begin{array}{rrr}{1} & {2} & {-1} \\ {0} & {3} & {1} \\ {2} & {-1} & {-2}\end{array}\right] \quad E=\left[\begin{array}{rrr}{2} & {-5} & {0} \\\ {1} & {0} & {-2} \\ {3} & {1} & {1}\end{array}\right] \quad F=\left[\begin{array}{rr}{-3} & {2} \\ {-5} & {1} \\ {2} & {4}\end{array}\right]$$ $$ (D E) F $$

Find the constant of variation for a direct variation that includes the given values. \((-4,-10)\)