Chapter 5

(a) write a matrix equation that is equivalent to the system of linear equations and (b) solve the system using the inverses found in Exercises 5-16. \(\begin{aligned} 2 x-3 y-4 z &=4 \\\\-z &=3 \\ x-2 y+z &=-8 \end{aligned}\) (See Exercise 9.)

Compute the indicated products. \(\left[\begin{array}{rrrr}3 & 0 & -2 & 1 \\ 1 & 2 & 0 & -1\end{array}\right]\left[\begin{array}{rrr}2 & 1 & -1 \\ -1 & 2 & 0 \\ 0 & 0 & 1 \\ -1 & -2 & 2\end{array}\right]\)

Perform the indicated operations. \(\frac{1}{2}\left[\begin{array}{rrrr}1 & 0 & 0 & -4 \\ 3 & 0 & -1 & 6 \\ -2 & 1 & -4 & 2\end{array}\right]+\frac{4}{3}\left[\begin{array}{rrrr}3 & 0 & -1 & 4 \\ -2 & 1 & -6 & 2 \\ 8 & 2 & 0 & -2\end{array}\right]\) \(-\frac{1}{3}\left[\begin{array}{rrrr}3 & -9 & -1 & 0 \\ 6 & 2 & 0 & -6 \\ 0 & 1 & -3 & 1\end{array}\right]\)

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{array}{rr}x-2 y= & 2 \\ 7 x-14 y= & 14 \\ 3 x-6 y= & 6\end{array}\)

Pivot the system about the circled element. \(\left[\begin{array}{cc|c}(2) & 4 & 8 \\ 3 & 1 & 2\end{array}\right]\)

Formulate but do not solve the problem. You will be asked to solve these problems in the next section. The total number of passengers riding a certain city bus during the morning shift is 1000 . If the child's fare is $$\$ .50$$, the adult fare is $$\$ 1.50$$, and the total revenue from the fares in the morning shift is $$\$ 1300$$, how many children and how many adults rode the bus during the morning shift?

Compute the indicated products. \(\left[\begin{array}{rrrr}2 & 1 & -3 & 0 \\ 4 & -2 & -1 & 1 \\ -1 & 2 & 0 & 1\end{array}\right]\left[\begin{array}{rr}2 & -1 \\ 1 & 4 \\ 3 & -3 \\ 0 & -5\end{array}\right]\)

Perform the indicated operations. \(0.5\left[\begin{array}{rrr}1 & 3 & 5 \\ 5 & 2 & -1 \\ -2 & 0 & 1\end{array}\right]-0.2\left[\begin{array}{rrr}2 & 3 & 4 \\ -1 & 1 & -4 \\ 3 & 5 & -5\end{array}\right]\) \(+0.6\left[\begin{array}{rrr}3 & 4 & -1 \\ 4 & 5 & 1 \\ 1 & 0 & 0\end{array}\right]\)

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{aligned} x+2 y+z &=-2 \\\\-2 x-3 y-z &=1 \\ 2 x+4 y+2 z &=-4 \end{aligned}\)

Pivot the system about the circled element. \(\left[\begin{array}{ll|l}3 & 2 & 6 \\ (4) & 2 & 5\end{array}\right]\)

Formulate but do not solve the problem. You will be asked to solve these problems in the next section. Cantwell Associates, a real estate developer, is planning to build a new apartment complex consisting of one-bedroom units and two- and three-bedroom townhouses. A total of 192 units is planned, and the number of family units (two- and three-bedroom townhouses) will equal the number of one-bedroom units. If the number of one-bedroom units will be 3 times the number of threebedroom units, find how many units of each type will be in the complex.

Compute the indicated products. \(4\left[\begin{array}{rrr}1 & -2 & 0 \\ 2 & -1 & 1 \\ 3 & 0 & -1\end{array}\right]\left[\begin{array}{rrr}1 & 3 & 1 \\ 1 & 4 & 0 \\ 0 & 1 & -2\end{array}\right]\)

Solve for \(u, x, y\), and \(z\) in the given matrix equation. \(\left[\begin{array}{crc}2 x-2 & 3 & 2 \\ 2 & 4 & y-2 \\ 2 z & -3 & 2\end{array}\right]=\left[\begin{array}{rrr}3 & u & 2 \\ 2 & 4 & 5 \\ 4 & -3 & 2\end{array}\right]\)

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

Formulate but do not solve the problem. You will be asked to solve these problems in the next section. he annual returns on Sid Carrington's three investments amounted to $$\$ 21,600: 6 \%$$ on a savings account, $$8 \%$$ on mutual funds, and $$12 \%$$ on bonds. The amount of Sid's investment in bonds was twice the amount of his investment in the savings account, and the interest earned from his investment in bonds was equal to the dividends he received from his investment in mutual funds. Find how much money he placed in each type of investment.

Compute the indicated products. \(3\left[\begin{array}{rrr}2 & -1 & 0 \\ 2 & 1 & 2 \\ 1 & 0 & -1\end{array}\right]\left[\begin{array}{rrr}2 & 3 & 1 \\ 3 & -3 & 0 \\ 0 & 1 & -1\end{array}\right]\)

Solve for \(u, x, y\), and \(z\) in the given matrix equation. \(\left[\begin{array}{rr}x & -2 \\ 3 & y\end{array}\right]+\left[\begin{array}{ll}-2 & z \\ -1 & 2\end{array}\right]=\left[\begin{array}{cr}4 & -2 \\ 2 u & 4\end{array}\right]\)

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

Pivot the system about the circled element. \(\left[\begin{array}{ll|l}(1) & 3 & 4 \\ 2 & 4 & 6\end{array}\right]\)

Formulate but do not solve the problem. You will be asked to solve these problems in the next section. A private investment club has $$\$ 200,000$$ earmarked for investment in stocks. To arrive at an acceptable overall level of risk, the stocks that management is considering have been classified into three categories: high risk, medium risk, and low risk. Management estimates that high-risk stocks will have a rate of return of $$15 \% / y e a r ;$$ medium-risk stocks, $$10 \% / y e a r ;$$ and low-risk stocks, $$6 \% /$$ year. The members have decided that the investment in low-risk stocks should be equal to the sum of the investments in the stocks of the other two categories. Determine how much the club should invest in each type of stock if the investment goal is to have a return of $$\$ 20,000 /$$ year on the total investment. (Assume that all the money available for investment is invested.)

Compute the indicated products. \(\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{rrr}4 & -3 & 2 \\ 7 & 1 & -5\end{array}\right]\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\)

Solve for \(u, x, y\), and \(z\) in the given matrix equation. \(\left[\begin{array}{rr}1 & x \\ 2 y & -3\end{array}\right]-4\left[\begin{array}{rr}2 & -2 \\ 0 & 3\end{array}\right]=\left[\begin{array}{cc}3 z & 10 \\ 4 & -u\end{array}\right]\)

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{aligned} 2 x_{1}-x_{2}+x_{3} &=-4 \\ 3 x_{1}-\frac{3}{2} x_{2}+\frac{3}{2} x_{3} &=-6 \\\\-6 x_{1}+3 x_{2}-3 x_{3} &=12 \end{aligned}\)

Pivot the system about the circled element. \(\left[\begin{array}{rrr|r}(2) & 4 & 6 & 12 \\ 2 & 3 & 1 & 5 \\ 3 & -1 & 2 & 4\end{array}\right]\)

Formulate but do not solve the problem. You will be asked to solve these problems in the next section. Lawnco produces three grades of commercial fertilizers. A 100 -lb bag of grade-A fertilizer contains 18 lb of nitrogen, 4 lb of phosphate, and 5 lb of potassium. A \(100-\mathrm{lb}\) bag of grade-B fertilizer contains \(20 \mathrm{lb}\) of nitrogen and \(4 \mathrm{lb}\) each of phosphate and potassium. A \(100-\mathrm{lb}\) bag of grade-C fertilizer contains 24 lb of nitrogen, 3 lb of phosphate, and 6 lb of potassium. How many \(100-\mathrm{lb}\) bags of each of the three grades of fertilizers should Lawnco produce if 26,400 lb of nitrogen, 4900 lb of phosphate, and \(6200 \mathrm{lb}\) of potassium are available and all the nutrients are used?

Compute the indicated products. \(2\left[\begin{array}{rrr}3 & 2 & -1 \\ 0 & 1 & 3 \\ 2 & 0 & 3\end{array}\right]\left[\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\left[\begin{array}{rrr}1 & 2 & 0 \\ 0 & -1 & -2 \\ 1 & 3 & 1\end{array}\right]\)

Solve for \(u, x, y\), and \(z\) in the given matrix equation. \(\left[\begin{array}{rr}1 & 2 \\ 3 & 4 \\ x & -1\end{array}\right]-3\left[\begin{array}{cc}y-1 & 2 \\ 1 & 2 \\ 4 & 2 z+1\end{array}\right]=2\left[\begin{array}{rr}-4 & -u \\ 0 & -1 \\ 4 & 4\end{array}\right]\)

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{array}{rr}x+y-2 z= & -3 \\ 2 x-y+3 z= & 7 \\ x-2 y+5 z= & 0\end{array}\)

Pivot the system about the circled element. \(\left[\begin{array}{rrr|r}1 & 3 & 2 & 4 \\ 2 & 4 & 8 & 6 \\ -1 & 2 & 3 & 4\end{array}\right]\)

(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. \(\quad x+2 y=b_{1}$$2 x-y=b_{2}\) where (i) \(b_{1}=14, b_{2}=5\) and (ii) \(b_{1}=4, b_{2}=-1\)

Let $$\begin{array}{l}A=\left[\begin{array}{rrr}1 & 0 & -2 \\\1 & -3 & 2 \\\\-2 & 1 & 1 \end{array}\right] \quad B=\left[\begin{array}{rrr}3 & 1 & 0 \\\2 & 2 & 0 \\ 1 & -3 & -1\end{array}\right] \\\C=\left[\begin{array}{lll}2 & -1 & 0 \\\1 & -1 & 2 \\\3 & -2 & 1\end{array}\right]\end{array}$$ Verify the validity of the associative law for matrix multiplication.

Let $$\begin{array}{l}A=\left[\begin{array}{rrr}2 & -4 & 3 \\\4 & 2 & 1\end{array}\right] \quad B=\left[\begin{array}{rrr}4 & -3 & 2 \\\1 & 0 & 4\end{array}\right] \\ C=\left[\begin{array}{rrr}1 & 0 & 2 \\\3 & -2 & 1\end{array}\right]\end{array}$$ Verify by direct computation the validity of the commutative law for matrix addition.

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

Pivot the system about the circled element. \(\left[\begin{array}{rrr|r}0 & 1 & 3 & 4 \\ 2 & 4 & 1 & 3 \\ 5 & 6 & 2 & -4\end{array}\right]\)

Formulate but do not solve the problem. You will be asked to solve these problems in the next section. The management of Hartman Rent-A-Car has allocated $$\$ 1.5$$ million to buy a fleet of new automobiles consisting of compact, intermediate-size, and full- size cars. Compacts cost $$\ 12,000$$ each, intermediate size cars cost $$\ 18,000$$ each, and full-size cars cost $$\ 24,000$$ each. If Hartman purchases twice as many compacts as intermediate-size cars and the total number of cars to be purchased is 100, determine how many cars of each type will be purchased. (Assume that the entire budget will be used.)

(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. \(3 x-2 y=b_{1}\) \(4 x+3 y=b_{2}\) where (i) \(b_{1}=-6, b_{2}=10\) and (ii) \(b_{1}=3, b_{2}=-2\)

Let $$\begin{array}{l}A=\left[\begin{array}{rrr}1 & 0 & -2 \\\1 & -3 & 2 \\\\-2 & 1 & 1 \end{array}\right] \quad B=\left[\begin{array}{rrr}3 & 1 & 0 \\\2 & 2 & 0 \\ 1 & -3 & -1\end{array}\right] \\\C=\left[\begin{array}{lll}2 & -1 & 0 \\\1 & -1 & 2 \\\3 & -2 & 1\end{array}\right]\end{array}$$ Verify the validity of the distributive law for matrix multiplication.

Let $$\begin{array}{l}A=\left[\begin{array}{rrr}2 & -4 & 3 \\\4 & 2 & 1\end{array}\right] \quad B=\left[\begin{array}{rrr}4 & -3 & 2 \\\1 & 0 & 4\end{array}\right] \\ C=\left[\begin{array}{rrr}1 & 0 & 2 \\\3 & -2 & 1\end{array}\right]\end{array}$$ Verify by direct computation the validity of the associative law for matrix addition.

Solve the system of linear equations, using the Gauss-Jordan elimination method. \(\begin{aligned} x_{1}-2 x_{2}+x_{3} &=-3 \\ 2 x_{1}+x_{2}-2 x_{3} &=2 \\\ x_{1}+3 x_{2}-3 x_{3} &=5 \end{aligned}\)

Pivot the system about the circled element. \(\left[\begin{array}{rrr|r}1 & 2 & 3 & 5 \\ 0 & -3 & 3 & 2 \\ 0 & 4 & -1 & 3\end{array}\right]\)

Formulate but do not solve the problem. You will be asked to solve these problems in the next section. The management of a private investment club has a fund of $$\$ 200,000$$ earmarked for investment in stocks. To arrive at an acceptable overall level of risk, the stocks that management is considering have been classified into three categories: high risk, medium risk, and low risk. Management estimates that high-risk stocks will have a rate of return of $$15 \% /$$ year; medium- risk stocks, $$10 \% /$$ year; and low-risk stocks, $$6 \% /$$ year. The investment in low-risk stocks is to be twice the sum of the investments in stocks of the other two categories. If the investment goal is to have an average rate of return of $$9 \% /$$ year on the total investment, determine how much the club should invest in each type of stock. (Assume that all the money available for investment is invested.)

(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. \(\begin{aligned} x+2 y+z &=b_{1} \\ x+y+z &=b_{2} \\ 3 x+y+z &=b_{3} \\\ \text { where } & \text { (i) } b_{1}=7, b_{2}=4, b_{3}=2 \\ \text { and } & \text { (ii) } b_{1}=5, b_{2}=-3, b_{3}=-1 \end{aligned}\)

Let $$A=\left[\begin{array}{ll}1 & 2 \\\3 & 4 \end{array}\right] \text { and } B=\left[\begin{array}{ll}2 & 1 \\\4 & 3\end{array}\right]$$ Compute \(A B\) and \(B A\) and hence deduce that matrix multiplication is, in general, not commutative.

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\\. \((3+5) A=3 A+5 A\)

Fill in the missing entries by performing the indicated row operations to obtain the row-reduced matrices. \(\left[\begin{array}{ll|l}3 & 9 & 6 \\ 2 & 1 & 4\end{array}\right] \stackrel{\downarrow R_{1}}{\longrightarrow}\left[\begin{array}{ll|l}\cdot & \cdot & \cdot \\ 2 & 1 & 4\end{array}\right] \frac{R_{2}-2 R_{1}}{\longrightarrow}\) \(\left[\begin{array}{ll|l}1 & 3 & 2 \\ \cdot & \cdot & \cdot\end{array}\right]-\frac{1}{3} R_{2}\left[\begin{array}{ll|l}1 & 3 & 2 \\\ \cdot & \cdot & \cdot\end{array}\right] \frac{R_{1}-3 R_{2}}{\longrightarrow}\left[\begin{array}{ll|l}1 & 0 & 2 \\ 0 & 1 & 0\end{array}\right]\)

(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. \(x_{1}+x_{2}+x_{3}=b_{1}\) \(x_{1}-x_{2}+x_{3}=b_{2}\) \(x_{1}-2 x_{2}-x_{3}=b_{3}\) where \(\quad\) (i) \(b_{1}=5, b_{2}=-3, b_{3}=-1\) and \(\quad\) (ii) \(b_{1}=1, b_{2}=4, b_{3}=-2\)

Let $$\begin{array}{l}A=\left[\begin{array}{lll}0 & 3 & 0 \\\1 & 0 & 1 \\\0 & 2 & 0\end{array}\right] \quad B=\left[\begin{array}{rrr}2 & 4 & 5 \\\3 & -1 & -6 \\\4 & 3 & 4\end{array}\right] \\\C=\left[\begin{array}{rrr} 4 & 5 & 6 \\\3 & -1 & -6 \\\2 & 2 & 3\end{array}\right]\end{array}$$ a. Compute \(\overline{A B}\). b. Compute \(A \bar{C}\). c. Using the results of parts (a) and (b), conclude that \(A B=A C\) does not imply that \(B=C\).

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\\. \(2(4 A)=(2 \cdot 4) A=8 A\)

Fill in the missing entries by performing the indicated row operations to obtain the row-reduced matrices. \(\left[\begin{array}{rr|r}1 & 2 & 1 \\ 2 & 3 & -1\end{array}\right] \stackrel{R_{2}-2 R_{1}}{\longrightarrow}\left[\begin{array}{ll|l}1 & 2 & 1 \\\ . & . & .\end{array}\right] \frac{-R_{2}}{\longrightarrow}\) \(\left[\begin{array}{ll|l}1 & 2 & 1 \\ . & . & .\end{array}\right] \frac{R_{1}-2 R_{2}}{\longrightarrow}\left[\begin{array}{ll|r}1 & 0 & -5 \\ 0 & 1 & 3\end{array}\right]\)

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. A system composed of two linear equations must have at least one solution if the straight lines represented by these equations are nonparallel.