Chapter 5

(a) write each system of equations as a matrix equation and (b) solve the system of equations by using the inverse of the coefficient matrix. \(3 x+2 y-z=b_{1}\) \(2 x-3 y+z=b_{2}\) \(x-y-z=b_{3}\) where \(\quad\) (i) \(b_{1}=2, b_{2}=-2, b_{3}=4\) and \(\quad\) (ii) \(b_{1}=8, b_{2}=-3, b_{3}=6\)

Let $$A=\left[\begin{array}{ll}3 & 0 \\\8 & 0\end{array}\right] \text { and } B=\left[\begin{array}{ll}0 & 0 \\\4 & 5\end{array}\right]$$ Show that \(A B=0\), thereby demonstrating that for matrix multiplication the equation \(A B=0\) does not imply that one or both of the matrices \(A\) and \(B\) must be the zero matrix.

Let $$A=\left[\begin{array}{rr}3 & 1 \\\2 & 4 \\\\-4 & 0\end{array}\right] \text { and } B=\left[\begin{array}{rr}1 & 2 \\\\-1 & 0 \\\3 & 2\end{array}\right]$$ Verify each equation by direct computation\\. \(4(A+B)=4 A+4 B\)

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(2 x+y-3 z=1\) \(x-y+2 z=1\) \(5 x-2 y+3 z=6\)

Fill in the missing entries by performing the indicated row operations to obtain the row-reduced matrices. \(\left[\begin{array}{rrr|r}1 & 3 & 1 & 3 \\ 3 & 8 & 3 & 7 \\ 2 & -3 & 1 & -10\end{array}\right] \frac{R_{2}-3 R_{1}}{R_{3}-2 R_{1}}\left[\begin{array}{ccc|c}1 & 3 & 1 & 3 \\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot\end{array}\right] \stackrel{-R_{2}}{\longrightarrow}\) \(\left[\begin{array}{ccc|c}1 & 3 & 1 & 3 \\ \cdot & \cdot & \cdot & \cdot \\\ 0 & -9 & -1 & -16\end{array}\right] \frac{R_{1}-3 R_{2}}{R_{3}+9 R_{2}}\) \(\left[\begin{array}{lll|l}\cdot & \cdot & \cdot & \cdot \\ 0 & 1 & 0 & 2 \\\ \cdot & \cdot & \cdot & \cdot\end{array}\right] \frac{R_{1}+R_{3}}{-R_{3}}\left[\begin{array}{lll|r}1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & -2\end{array}\right]\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. Suppose the straight lines represented by a system of three linear equations in two variables are parallel to each other. Then the system has no solution or it has infinitely many solutions.

(a) write each system of equations as a matrix equation and (b) solve the system of equations by using the inverse of the coefficient matrix. \(\begin{aligned} 2 x_{1}+x_{2}+x_{3}=b_{1} \\ x_{1}-3 x_{2}+4 x_{3}=b_{2} \\\\-x_{1}+x_{3}=b_{3} \\ \text { where } & \text { (i) } b_{1}=1, b_{2}=4, b_{3}=-3 \\ \text { and } & \text { (ii) } b_{1}=2, b_{2}=-5, b_{3}=0 \end{aligned}\)

Let $$A=\left[\begin{array}{rr}2 & 2 \\\\-2 & -2\end{array}\right]$$ Show that \(A^{2}=0\). Compare this with the equation \(a^{2}=0\), where \(a\) is a real number.

Let $$A=\left[\begin{array}{rr}3 & 1 \\\2 & 4 \\\\-4 & 0\end{array}\right] \text { and } B=\left[\begin{array}{rr}1 & 2 \\\\-1 & 0 \\\3 & 2\end{array}\right]$$ Verify each equation by direct computation\\. \(2(A-3 B)=2 A-6 B\)

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{array}{rr}3 x-9 y+6 z= & -12 \\ x-3 y+2 z= & -4 \\ 2 x-6 y+4 z= & 8\end{array}\)

Fill in the missing entries by performing the indicated row operations to obtain the row-reduced matrices. \(\left[\begin{array}{rrr|r}0 & 1 & 3 & -4 \\ 1 & 2 & 1 & 7 \\ 1 & -2 & 0 & 1\end{array}\right] \frac{R_{1} \leftrightarrow R_{2}}{\longrightarrow}\left[\begin{array}{rrr|r}. & \cdot & \cdot & \cdot \\\ \cdot & \cdot & \cdot & \cdot \\ 1 & -2 & 0 & 1\end{array}\right]\) \(\frac{R_{3}-R_{1}}{\longrightarrow}\left[\begin{array}{rrr|r}1 & 2 & 1 & 7 \\\ 0 & 1 & 3 & -4 \\ \cdot & \cdot & \cdot & \cdot\end{array}\right] \frac{R_{1}+\frac{1}{2} R_{3}}{R_{3}+4 R_{2}}\left[\begin{array}{ccc|r}\cdot & \cdot & \cdot & \cdot \\ 0 & 1 & 3 & -4 \\ \cdot & \cdot & \cdot & \cdot\end{array}\right]\)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If at least two of the three lines represented by a system composed of three linear equations in two variables are parallel, then the system has no solution.

(a) write each system of equations as a matrix equation and (b) solve the system of equations by using the inverse of the coefficient matrix. \(x_{1}+x_{2}+x_{3}+x_{4}=b_{1}\) \(x_{1}-x_{2}-x_{3}+x_{4}=b_{2}\) \(x_{2}+2 x_{3}+2 x_{4}=b_{3}\) \(x_{1}+2 x_{2}+x_{3}-2 x_{4}=b_{4}\) where (i) \(b_{1}=1, b_{2}=-1, b_{3}=4, b_{4}=0\) and (ii) \(b_{1}=2, b_{2}=8, b_{3}=4, b_{4}=-1\)

Find the matrix \(A\) such that $$A\left[\begin{array}{rr}1 & 0 \\\\-1 & 3\end{array}\right]=\left[\begin{array}{rr}-1 & -3 \\\3 & 6\end{array}\right]$$

Find the transpose of each matrix. \(\left[\begin{array}{llll}3 & 2 & -1 & 5\end{array}\right]\)

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{array}{rr}x+2 y-z= & -4 \\ 2 x+y+z= & 7 \\ x+3 y+2 z= & 7 \\ x-3 y+z= & 9\end{array}\)

(a) write each system of equations as a matrix equation and (b) solve the system of equations by using the inverse of the coefficient matrix. \(\begin{aligned} x_{1}+x_{2}+2 x_{3}+x_{4} &=b_{1} \\ 4 x_{1}+5 x_{2}+9 x_{3}+x_{4} &=b_{2} \\ 3 x_{1}+4 x_{2}+7 x_{3}+x_{4} &=b_{3} \\ 2 x_{1}+3 x_{2}+4 x_{3}+2 x_{4} &=b_{4} \end{aligned}\) where (i) \(b_{1}=3, b_{2}=6, b_{3}=5, b_{4}=7\) and (ii) \(b_{1}=1, b_{2}=-1, b_{3}=0, b_{4}=-4\)

Let $$A=\left[\begin{array}{rr}2 & 4 \\\5 & -6\end{array}\right] \text { and } B=\left[\begin{array}{rr}4 & 8 \\ -7 & 3\end{array}\right]$$ a. Find \(A^{T}\) and show that \(\left(A^{T}\right)^{T}=A\). b. Show that \((A+B)^{T}=A^{T}+B^{T}\). c. Show that \((A B)^{T}=B^{T} A^{T}\).

Find the transpose of each matrix. \(\left[\begin{array}{rrrr}4 & 2 & 0 & -1 \\ 3 & 4 & -1 & 5\end{array}\right]\)

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{array}{rr}3 x-2 y+z= & 4 \\ x+3 y-4 z= & -3 \\ 2 x-3 y+5 z= & 7 \\\ x-8 y+9 z= & 10\end{array}\)

Let $$A=\left[\begin{array}{rr}2 & 3 \\\\-4 & -5\end{array}\right]$$ a. Find \(A^{-1}\). b. Show that \(\left(A^{-1}\right)^{-1}=A\).

Let $$A=\left[\begin{array}{rr}2 & 4 \\\5 & -6\end{array}\right] \text { and } B=\left[\begin{array}{rr}4 & 8 \\ -7 & 3\end{array}\right]$$ a. Find \(A^{T}\) and show that \(\left(A^{T}\right)^{T}=A\). b. Show that \((A+B)^{T}=A^{T}+B^{T}\). c. Show that \((A B)^{T}=B^{T} A^{T}\).

Find the transpose of each matrix. \(\left[\begin{array}{rrr}1 & -1 & 2 \\ 3 & 4 & 2 \\ 0 & 1 & 0\end{array}\right]\)

The management of Hartman Rent-A-Car has allocated $$\$ 1,008,000$$ to purchase 60 new automobiles to add to their existing fleet of rental cars. The company will choose from compact, mid-sized, and fullsized cars costing \(\$ 12,000, \$ 19,200\), and \(\$ 26,400\) each, respectively. Find formulas giving the options available to the company. Give two specific options. (Note: Your answers will not be unique.)

Let $$A=\left[\begin{array}{rr}6 & -4 \\ -4 & 3\end{array}\right] \text { and } \quad B=\left[\begin{array}{ll} 3 & -5 \\\4 & -7\end{array}\right]$$ a. Find \(A B, A^{-1}\), and \(B^{-1}\). b. Show that \((A B)^{-1}=B^{-1} A^{-1}\).

Let $$A=\left[\begin{array}{rr}1 & 3 \\\\-2 & -1\end{array}\right] \text { and } B=\left[\begin{array}{ll}3 & -4 \\\2 & -2 \end{array}\right]$$ a. Find \(A^{T}\) and show that \(\left(A^{T}\right)^{T}=A\). b. Show that \((A+B)^{T}=A^{T}+B^{T}\). c. Show that \((A B)^{T}=B^{T} A^{T}\).

Find the transpose of each matrix. \(\left[\begin{array}{llll}1 & 2 & 6 & 4 \\ 2 & 3 & 2 & 5 \\ 6 & 2 & 3 & 0 \\\ 4 & 5 & 0 & 2\end{array}\right]\)

A dietitian wishes to plan a meal around three foods. The meal is to include 8800 units of vitamin A, 3380 units of vitamin \(\mathrm{C}\), and 1020 units of calcium. The number of units of the vitamins and calcium in each ounce of the foods is summarized in the following table: $$\begin{array}{lccc}\hline & \text { Food I } & \text { Food II } & \text { Food III } \\\\\hline \text { Vitamin A } & 400 & 1200 & 800 \\\\\hline \text { Vitamin C } & 110 & 570 & 340 \\\\\hline \text { Calcium } & 90 & 30 & 60 \\\\\hline\end{array}$$ Determine the amount of each food the dietitian should include in the meal in order to meet the vitamin and calcium requirements.

Let \(A=\left[\begin{array}{ll}2 & -5 \\ 1 & -3\end{array}\right] \quad B=\left[\begin{array}{ll}4 & 3 \\ 1 & 1\end{array}\right] \quad C=\left[\begin{array}{rr}2 & 3 \\ -2 & 1\end{array}\right]\) a. Find \(A B C, A^{-1}, B^{-1}\), and \(C^{-1}\). b. Show that \((A B C)^{-1}=C^{-1} B^{-1} A^{-1}\).

Write the given system of linear equations in matrix form. \(\begin{aligned} 2 x-3 y &=7 \\ 3 x-4 y &=8 \end{aligned}\)

Mr. Cross, Mr. Jones, and Mr. Smith each suffer from coronary heart disease. As part of their treatment, they were put on special low-cholesterol diets: Cross on diet I, Jones on diet II, and Smith on diet III. Progressive records of each patient's cholesterol level were kept. At the beginning of the first, second, third, and fourth months, the cholesterol levels of the three patients were: Cross: \(220,215,210\), and 205 Jones: \(220,210,200\), and 195 Smith: \(215,205,195\), and 190 Represent this information in a \(3 \times 4\) matrix.

Solve the system of linear equations using the Gauss-Jordan elimination method. \(\begin{aligned} x-2 y &=8 \\ 3 x+4 y &=4 \end{aligned}\)

Find the matrix \(A\) if $$\left[\begin{array}{rr}2 & 2 \\\\-1 & 3\end{array}\right] A=\left[\begin{array}{ll}3 & 2 \\ 1 & 4\end{array}\right]$$

Write the given system of linear equations in matrix form. \(\begin{aligned} 2 x &=7 \\ 3 x-2 y &=12 \end{aligned}\)

The following table gives the number of shares of certain corporations held by Leslie and Tom in their respective IRA accounts at the beginning of the year: $$\begin{array}{lcccc}\hline & \text { IBM } & \text { GE } & \text { Ford } & \text { Wal-Mart } \\ \hline \text { Leslie } & 500 & 350 & 200 & 400 \\\\\hline \text { Tom } & 400 & 450 & 300 & 200 \\ \hline\end{array}$$ Over the year, they added more shares to their accounts, as shown in the following table: a. Write a matrix \(A\) giving the holdings of Leslie and Tom at the beginning of the year and a matrix \(B\) giving the shares they have added to their portfolios. b. Find a matrix \(C\) giving their total holdings at the end of the year. $$\begin{array}{lcccc}\hline & \text { IBM } & \text { GE } & \text { Ford } & \text { Wal-Mart } \\\\\hline \text { Leslie } & 50 & 50 & 0 & 100 \\\\\hline \text { Tom } & 0 & 80 & 100 & 50 \\\\\hline\end{array}$$ a. Write a matrix \(A\) giving the holdings of Leslie and Tom at the beginning of the year and a matrix \(B\) giving the shares they have added to their portfolios. b. Find a matrix \(C\) giving their total holdings at the end of the year.

Solve the system of linear equations using the Gauss-Jordan elimination method. \(\begin{aligned} 3 x+y &=1 \\\\-7 x-2 y &=-1 \end{aligned}\)

Find the matrix \(A\) if $$A\left[\begin{array}{rr}1 & 2 \\ 3 & -1\end{array}\right]=\left[\begin{array}{rr}2 & 1 \\\3 & -2\end{array}\right]$$

Write the given system of linear equations in matrix form. \(\begin{aligned} 2 x-3 y+4 z &=6 \\ 2 y-3 z &=7 \\ x-y+2 z &=4 \end{aligned}\)

Builders build three models of houses, \(M_{1}, M_{2}\), and \(M_{3}\), in three subdivisions I, II, and III located in three different areas of a city. The prices of the houses (in thousands of dollars) are given in matrix \(A\) : \(\mathrm{K}\) \& R Builders has decided to raise the price of each house by \(3 \%\) next year. Write a matrix \(B\) giving the new prices of the houses.

Mr. and Mrs. Garcia have a total of $$\$ 100,000$$ to be invested in stocks, bonds, and a money market account. The stocks have a rate of return of \(12 \%\) lyear, while the bonds and the money market account pay \(8 \% /\) year and \(4 \%\) year, respectively. The Garcias have stipulated that the amount invested in stocks should be equal to the sum of the amount invested in bonds and 3 times the amount invested in the money market account. How should the Garcias allocate their resources if they require an annual income of \(\$ 10,000\) from their investments? Give two specific options.

Solve the system of linear equations using the Gauss-Jordan elimination method. \(\begin{aligned} 2 x-3 y &=-8 \\ 4 x+y &=-2 \end{aligned}\)

Rainbow Harbor Cruises charges \$16/adult and \(\$ 8 /\) child for a round-trip ticket. The records show that, on a certain weekend, 1000 people took the cruise on Saturday and 800 people took the cruise on Sunday. The total receipts for Saturday were \(\$ 12,800\) and the total receipts for Sunday were \(\$ 9,600\). Determine how many adults and children took the cruise on Saturday and on Sunday.

Write the given system of linear equations in matrix form. \(\begin{aligned} x-2 y+3 z=&-1 \\ 3 x+4 y-2 z=& 1 \\ 2 x-3 y+7 z=& 6 \end{aligned}\)

Solve the system of linear equations using the Gauss-Jordan elimination method. \(\begin{aligned} 5 x+3 y &=9 \\\\-2 x+y &=-8 \end{aligned}\)

Publishing publishes a deluxe leather edition and a standard edition of its Daily Organizer. The company's marketing department estimates that \(x\) copies of the deluxe edition and \(y\) copies of the standard edition will be demanded per month when the unit prices are \(p\) dollars and \(q\) dollars, respectively, where \(x, y, p\), and \(q\) are related by the following system of linear equations: $$\begin{aligned}5 x+y &=1000(70-p) \\\x+3 y &=1000(40-q)\end{aligned}$$ Find the monthly demand for the deluxe edition and the standard edition when the unit prices are set according to the following schedules: a. \(p=50\) and \(q=25\) b. \(p=45\) and \(q=25\) c. \(p=45\) and \(q=20\)

Write the given system of linear equations in matrix form. \(\begin{aligned}-x_{1}+x_{2}+x_{3} &=0 \\ 2 x_{1}-x_{2}-x_{3} &=2 \\\\-3 x_{1}+2 x_{2}+4 x_{3} &=4 \end{aligned}\)

The numbers of three types of bank accounts on January 1 at the Central Bank and its branches are represented by matrix \(A\) : The number and types of accounts opened during the first quarter are represented by matrix \(B\), and the number and types of accounts closed during the same period are represented by matrix \(C\). Thus, \(B=\left[\begin{array}{rrr}260 & 120 & 110 \\ 140 & 60 & 50 \\ 120 & 70 & 50\end{array}\right]\) and \(C=\left[\begin{array}{rrr}120 & 80 & 80 \\ 70 & 30 & 40 \\ 60 & 20 & 40\end{array}\right]\) a. Find matrix \(D\), which represents the number of each type of account at the end of the first quarter at each location. b. Because a new manufacturing plant is opening in the immediate area, it is anticipated that there will be a \(10 \%\) increase in the number of accounts at each location during the second quarter. Write a matrix \(E=1.1 D\) to reflect this anticipated increase.

Solve the system of linear equations using the Gauss-Jordan elimination method. \(\begin{aligned} x+y+z &=0 \\ 2 x-y+z &=1 \\ x+y-2 z &=2 \end{aligned}\)

Bob, a nutritionist who works for the University Medical Center, has been asked to prepare special diets for two patients, Susan and Tom. Bob has decided that Susan's meals should contain at least \(400 \mathrm{mg}\) of calcium, \(20 \mathrm{mg}\) of iron, and \(50 \mathrm{mg}\) of vitamin \(\mathrm{C}\), whereas Tom's meals should contain at least \(350 \mathrm{mg}\) of calcium, \(15 \mathrm{mg}\) of iron, and \(40 \mathrm{mg}\) of vitamin \(\mathrm{C}\). Bob has also decided that the meals are to be prepared from three basic foods: food \(\mathrm{A}\), food \(\mathrm{B}\), and \(\mathrm{food} \mathrm{C}\). The special nutritional contents of these foods are summarized in the accompanying table. Find how many ounces of each type of food should be used in a meal so that the minimum requirements of calcium, iron, and vitamin \(\mathrm{C}\) are met for each patient's meals. $$\begin{array}{lccc}\hline & {\text { Contents (mg/oz) }} & \\ & \text { Calcium } & \text { Iron } & \text { Vitamin C } \\ \hline \text { Food A } & 30 & 1 & 2 \\ \hline \text { Food B } & 25 & 1 & 5 \\ \hline \text { Food C } & 20 & 2 & 4 \\ \hline\end{array}$$

Write the given system of linear equations in matrix form. \(\begin{aligned} 3 x_{1}-5 x_{2}+4 x_{3}=& 10 \\ 4 x_{1}+2 x_{2}-3 x_{3}=&-12 \\\\-x_{1} \quad+x_{3}=&-2 \end{aligned}\)