Chapter 9

If \(A=\left(a_{i j}\right)\) is a square matrix of order \(n\) and \(k\) is any real number, show that \(|k A|=k^{n}|A|\). (Hint: Use property 2 of the theorem on row and column transformations of a determinant.)

If \(A=\left(a_{i j}\right)\) is any \(2 \times 2\) matrix such that \(|A| \neq 0\), show that \(A\) has an inverse, and find a general formula for \(A^{-1}\).

Let \(A=\left[\begin{array}{rr}1 & 2 \\ 0 & -3\end{array}\right], \quad B=\left[\begin{array}{rr}2 & -1 \\ 3 & 1\end{array}\right], \quad C=\left[\begin{array}{rr}3 & 1 \\ -2 & 0\end{array}\right]\) $$ A(B C)=(A B) C $$

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{array}{rr} 2 x-3 y-z^{2}= & 0 \\ x-y-z^{2}= & -1 \\ x^{2}-x y= & 0 \end{array}\right. $$

A 300-gallon water storage tank is filled by a single inlet pipe, and two identical outlet pipes can be used to supply water to the surrounding fields (see the figure). It takes 5 hours to fill an empty tank when both outlet pipes are open. When one outlet pipe is closed, it takes 3 hours to fill the tank. Find the flow rates (in gallons per hour) in and out of the pipes.

Use properties of determinants to show that the following is an equation of a line through the points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) : $$ \left|\begin{array}{lll} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{array}\right|=0 $$

Verify the identity for $$ A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right], \quad B=\left[\begin{array}{ll} p & q \\ r & s \end{array}\right], \quad C=\left[\begin{array}{ll} w & x \\ y & z \end{array}\right] $$ and real numbers \(m\) and \(n\). $$ m(A+B)=m A+m B $$

Blending coffees A shop specializes in preparing blends of gourmet coffees. From Colombian, Costa Rican, and Kenyan coffees, the owner wishes to prepare 1-pound bags that will sell for \(\$ 12.50\). The cost per pound of these coffees is \(\$ 14, \$ 10\), and \(\$ 12\), respectively. The amount of Colombian is to be three times the amount of Costa Rican. Find the amount of each type of coffee in the blend.

$$ \left\\{\begin{array}{r} x^{2}+z^{2}=5 \\ 2 x+y=1 \\ y+z=1 \end{array}\right. $$

Mixing a silver alloy A silversmith has two alloys, one containing \(35 \%\) silver and the other \(60 \%\) silver. How much of each should be melted and combined to obtain 100 grams of an alloy containing \(50 \%\) silver?

Use properties of determinants to show that the following is an equation of a circle through three noncollinear points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\), and \(\left(x_{3}, y_{3}\right)\) : $$ \left|\begin{array}{llll} x^{2}+y^{2} & x & y & 1 \\ x_{1}^{2}+y_{1}^{2} & x_{1} & y_{1} & 1 \\ x_{2}^{2}+y_{2}^{2} & x_{2} & y_{2} & 1 \\ x_{3}^{2}+y_{3}^{2} & x_{3} & y_{3} & 1 \end{array}\right|=0 $$

Verify the identity for $$ A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right], \quad B=\left[\begin{array}{ll} p & q \\ r & s \end{array}\right], \quad C=\left[\begin{array}{ll} w & x \\ y & z \end{array}\right] $$ and real numbers \(m\) and \(n\). $$ (m+n) A=m A+n A $$

$$ \left\\{\begin{array}{r} x+2 z=1 \\ 2 y-z=4 \\ x y z=0 \end{array}\right. $$

A merchant wishes to mix peanuts costing $$\$ 3$$ per pound with cashews costing $$\$ 8$$ per pound to obtain 60 pounds of a mixture costing $$\$ 5$$ per pound. How many pounds of each variety should be mixed?

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{array}{r} 2 x+3 y=2 \\ x-2 y=8 \end{array}\right. $$

Verify the identity for $$ A=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right], \quad B=\left[\begin{array}{ll} p & q \\ r & s \end{array}\right], \quad C=\left[\begin{array}{ll} w & x \\ y & z \end{array}\right] $$ and real numbers \(m\) and \(n\). $$ A(B+C)=A B+A C $$

An airplane, flying with a tail wind, travels 1200 miles in 2 hours. The return trip, against the wind, takes \(2 \frac{1}{2}\) hours. Find the cruising speed of the plane and the speed of the wind (assume that both rates are constant).

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{array}{l} 4 x+5 y=13 \\ 3 x+y=-4 \end{array}\right. $$

If \(f(x)=a x^{3}+b x+c\), determine \(a, b\), and \(c\) such that the graph of \(f\) passes through the points \(P(-3,-12)\), \(Q(-1,22)\), and \(R(2,13)\)

Find the values of \(b\) such that the system $$ \left\\{\begin{aligned} x^{2}+y^{2} &=4 \\ y &=x+b \end{aligned}\right. $$ has (a) one solution (b) two solutions (c) no solution Interpret (a)-(c) graphically.

https://cdn.mathpix.com/snip/images/s-MwHVYwjmd9vrzH2VW4iSfM4laxim85MLTkv5d7GPw.original.fullsize.png

A stationery company sells two types of notepads to college bookstores, the first wholesaling for \(50 \not\) and the second for \(70 \notin\). The company receives an order for 500 notepads, together with a check for $$\$ 286$$. If the order fails to specify the number of each type, how should the company fill the order?

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{array}{l} 2 x+5 y=16 \\ 3 x-7 y=24 \end{array}\right. $$

Is there a real number \(x\) such that \(x=2^{-x}\) ? Decide by displaying graphically the system $$ \left\\{\begin{array}{l} y=x \\ y=2^{-x} \end{array}\right. $$

Inventory levels A store sells two brands of television sets. Customer demand indicates that it is necessary to stock at least twice as many sets of brand \(\mathrm{A}\) as of brand \(\mathrm{B}\). It is also necessary to have on hand at least 10 sets of brand \(B\). There is room for not more than 100 sets in the store. Find and graph a system of inequalities that describes all possibilities for stocking the two brands.

As a ball rolls down an inclined plane, its velocity \(v(t)\) (in \(\mathrm{cm} / \mathrm{sec}\) ) at time \(t\) (in seconds) is given by \(v(t)=v_{0}+a t\) for initial velocity \(v_{0}\) and acceleration \(a\) (in \(\left.\mathrm{cm} / \mathrm{sec}^{2}\right)\). If \(v(2)=16\) and \(v(5)=25\), find \(v_{0}\) and \(a\).

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{array}{l} 7 x-8 y=9 \\ 4 x+3 y=-10 \end{array}\right. $$

Building costs A housing contractor has orders for 4 onebedroom units, 10 two- bedroom units, and 6 three-bedroom units. The labor and material costs (in thousands of dollars) are given in the following table. $$ \begin{array}{|l|ccc|} \hline & \text { 1-Bedroom } & \text { 2-Bedroom } & \text { 3-Bedroom } \\ \hline \text { Labor } & 70 & 95 & 117 \\ \text { Materials } & 90 & 105 & 223 \\ \hline \end{array} $$ (a) Organize these data into an order matrix \(A\) and a cost matrix \(B\) so that the product \(C=A B\) is defined. (b) Find \(C\). (c) Interpret the meaning of each element in \(C\).

Is there a real number \(x\) such that \(x=\log x\) ? Decide by displaying graphically the system $$ \left\\{\begin{array}{l} y=x \\ y=\log x \end{array}\right. $$

Ticket prices An auditorium contains 600 seats. For an upcoming event, tickets will be priced at $$ 8\( for some seats and $$ 5\) for others. At least 225 tickets are to be priced at $$ 5\(, and total sales of at least $$ 3000\) are desired. Find and graph a system of inequalities that describes all possibilities for pricing the two types of tickets.

If an object is projected vertically upward from an altitude of \(s_{0}\) feet with an initial velocity of \(v_{0} \mathrm{ft} / \mathrm{sec}\), then its distance \(s(t)\) above the ground after \(t\) seconds is $$ s(t)=-16 t^{2}+v_{0} t+s_{0} . $$ If \(s(1)=84\) and \(s(2)=116\), what are \(v_{0}\) and \(s_{0}\) ?

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{aligned} 2 x-3 y &=5 \\ -6 x+9 y &=12 \end{aligned}\right. $$

Investment strategy A woman with \(\$ 15,000\) to invest decides to place at least \(\$ 2000\) in a high-risk, high-yield investment and at least three times that amount in a low-risk, low-yield investment. Find and graph a system of inequalities that describes all possibilities for placing the money in the two investments.

A small furniture company manufactures sofas and recliners. Each sofa requires 8 hours of labor and $$\$ 180$$ in materials, while a recliner can be built for $$\$ 105$$ in 6 hours. The company has 340 hours of labor available each week and can afford to buy $$\$ 6750$$ worth of materials. How many recliners and sofas can be produced if all labor hours and all materials must be used?

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{array}{r} 3 p-q=7 \\ -12 p+4 q=3 \end{array}\right. $$

A rancher is preparing an oat-cornmeal mixture for livestock. Each ounce of oats provides 4 grams of protein and 18 grams of carbohydrates, and an ounce of cornmeal provides 3 grams of protein and 24 grams of carbohydrates. How many ounces of each can be used to meet the nutritional goals of 200 grams of protein and 1320 grams of carbohydrates per feeding?

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{array}{rr} x-2 y-3 z= & -1 \\ 2 x+y+z= & 6 \\ x+3 y-2 z= & 13 \end{array}\right. $$

\(\left|\begin{array}{rrr}i & j & k \\ 2 & -1 & 6 \\ -3 & 5 & 1\end{array}\right|\)

Dimensions of a can An aerosol can is to be constructed in the shape of a circular cylinder with a small cone on the top. The total height of the can including the conical top is to be no more than 9 inches, and the cylinder must contain at least \(75 \%\) of the total volume. In addition, the height of the conical top must be at least 1 inch. Find and graph a system of inequalities that describes all possibilities for the relationship between the height \(y\) of the cylinder and the height \(x\) of the cone.

A plumber and an electrician are each doing repairs on their offices and agree to swap services. The number of hours spent on each of the projects is shown in the following table. $$ \begin{array}{|l|c|c|} \hline & \begin{array}{c} \text { Plumber's } \\ \text { office } \end{array} & \begin{array}{c} \text { Electrician's } \\ \text { office } \end{array} \\ \hline \text { Plumber's hours } & 6 & 4 \\ \text { Electrician's hours } & 5 & 6 \\ \hline \end{array} $$ They would prefer to call the matter even, but because of tax laws, they must charge for all work performed. They agree to select hourly wage rates so that the bill on each project will match the income that each person would ordinarily receive for a comparable job. (a) If \(x\) and \(y\) denote the hourly wages of the plumber and electrician, respectively, show that $$ 6 x+5 y=10 x \text { and } 4 x+6 y=11 y . $$ Describe the solutions to this system. (b) If the plumber ordinarily makes \(\$ 35\) per hour, what should the electrician charge?

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{array}{rr} x+3 y-z= & -3 \\ 3 x-y+2 z= & 1 \\ 2 x-y+z= & -1 \end{array}\right. $$

\(\left|\begin{array}{rrr}i & j & k \\ 1 & -2 & 3 \\ 2 & 1 & -4\end{array}\right|\)

Find equations for the altitudes of the triangle with vertices \(A(-3,2), B(5,4)\), and \(C(3,-8)\), and find the point at which the altitudes intersect.

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{array}{rr} 5 x+2 y-z= & -7 \\ x-2 y+2 z= & 0 \\ 3 y+z= & 17 \end{array}\right. $$

The perimeter of a rectangle is 40 inches, and its area is 96 in \(^{2}\). Find its length and width.

Locating a power plant A nuclear power plant will be constructed to serve the power needs of cities A and B. City B is 100 miles due east of \(\mathrm{A}\). The state has promised that the plant will be at least 60 miles from each city. It is not possible, however, to locate the plant south of either city because of rough terrain, and the plant must be within 100 miles of both \(A\) and \(B\). Assuming \(A\) is at the origin, find and graph a system of inequalities that describes all possible locations for the plant.

As a result of urbanization, the temperatures in Paris have increased. In 1891 the average daily minimum and maximum temperatures were \(5.8^{\circ} \mathrm{C}\) and \(15.1^{\circ} \mathrm{C}\), respectively. Between 1891 and 1968 , these average temperatures rose \(0.019^{\circ} \mathrm{C} / \mathrm{yr}\) and \(0.011^{\circ} \mathrm{C} / \mathrm{yr}\), respectively. Assuming the increases were linear, find the year when the difference between the minimum and maximum temperatures was \(9^{\circ} \mathrm{C}\), and determine the corresponding average maximum temperature.

Exer. 33-42: Use Cramer's rule, whenever applicable, to solve the system. $$ \left\\{\begin{aligned} 4 x-y+3 z &=6 \\ -8 x+3 y-5 z &=-6 \\ 5 x-4 y &=-9 \end{aligned}\right. $$

\(\left|\begin{array}{rrr}i & j & k \\ 4 & -6 & 2 \\ -2 & 3 & -1\end{array}\right|\)

A telephone company charges customers a certain amount for the first minute of a long distance call and another amount for each additional minute. A customer makes two calls to the same citya 36 -minute call for $$\$ 2.93$$ and a 13-minute call for $$\$ 1.09$$. (a) Determine the cost for the first minute and the cost for each additional minute. (b) If there is a federal tax rate of \(3.2 \%\) and a state tax rate of \(7.2 \%\) on all long distance calls, find, to the nearest minute, the longest call to the same city whose cost will not exceed \$5.00.