2.1 [35] 

Create a worksheet that you use to compute the following matrix products. Store each matrix factor and each product in its own portion of the worksheet.

(a) [5] 

Find the matrix product of A•B where A = and B = .

(b) [5] 

Can you find the matrix product B • A of the matrices in (a)? If you can, do it. If not, why not? If not, type your explanation of why not directly into the worksheet in a cell.

(c) [5] 

Find the matrix product C • D where C = and D = .

(d) [5] 

Verify that (C • D)^t = D^t • C^t using the matrices C and D of part (c). Express this verification by computing the cell-by-cell difference of the respective matrix elements, and showing that it's a matrix of zeros.

(e) [5] 

In parts (e), (f) and (g) of this problem, we explore a different way to express matrix multiplication. Define a named array MatrixE that contains and a named array MatrixF that contains . Define the names Array11 to be the first column of MatrixE, Array12 to be the second column of MatrixE, Array21 to be the first row of MatrixF, and Array22 to be the second row of MatrixF. Find the array product of Array11 and Array21. Also find the array product of Array12 and Array22. By subtracting the matrix product Array11 • Array21 from the array product of Array11 and Array21, verify that the array product of Array11 and Array21 is equal to the matrix product Array11 • Array21.

(f) [5] 

Find the sum of the two array products you found in (e).

(g) [5] 

Verify that the result in (f) is identical to the matrix product MatrixE • MatrixF. Express this verification by computing the cell-by-cell difference of the respective matrix elements, and showing that it's a matrix of zeros.

2.2 [40] 

Top management of Geodesic Telecommunications, Inc., has asked you to take a closer look at its launching plan (see Problem 1.1). Recall that they plan to build a global satellite network to provide wireless communications service to the world. To accomplish this, they plan to launch 480 communications satellites into low earth orbit.

In this problem, you'll examine costs from a perspective that differs from Problem 1.1. As in Problem 1.1, satellites are launched in groups of 8 by a single rocket, but not all satellites on a given rocket are identical. Some of the satellites are of a type called "Master Router" which handles communications between the satellites in the group, and some are "Soldier" satellites that carry the routine earth-orbit communications for customers. The Master Router also handles earth-orbit communications, but it sometimes happens that a customer on the ground who is connected to one satellite wants to communicate with another customer on the ground, who is connected to a different satellite. In this case, the two satellites contact the Master Router, which provides directions to them on how to route the communication.

There are six basic components needed to build all of the satellites. The components required to build the satellites are shown in the table below.

Component	Function	Master Router	Soldier	Cost (k$)
Relay Communicator	Transmit and receive between Soldiers and Master Router	6	1	144
Spare Relay Communicator	Spare above	2	1	144
Earth/Orbit Communicator	Transmit and receive between earth and a satellite	1	1	186
Relay Antennas	Antennas for communications between Soldiers and Master Router	5	1	79
Scheduling Equipment	Schedule conversations with Master Router	1	0	102
Base Unit	Chassis, power supply, attitude control, etc.	1	1	494

Starting with the fourth launch, three Master Router satellites are launched as part of the payload of every fourth rocket. The other five satellites on those rockets are Soldiers. All other satellites on all other rockets are Soldiers. Launches occur at the rate of three per week. Specifically, the first three launches are all Soldiers. The fourth launch is comprised of three Master Routers and five Soldiers. Thereafter, the same pattern of four launches repeats.

(a) [5] 

Compute the cumulative numbers of Master Routers and Soldiers in orbit at the end of each of the first 20 weeks. Display your result as a 2x20 range, with Master Routers in the top row, and Soldiers in the bottom row.

(b) [10] 

Compute the cumulative numbers of each of the six components required each week for the first 20 weeks. Your answer should be in the form of six rows of 20 numbers, one row for each of the six components.

(c) [10] 

Using the result of (b), compute the cumulative cost, in millions of dollars, of acquiring the six categories of components, by component category, by week. Also compute a weekly total. If you could not complete (a), create a row of 20 numbers, having the values 1 to 20, and use that instead. Your answer should be in the form of seven rows of 20 numbers, one row for each of the six components, and a seventh row at the bottom for the total.

(d) [15] 

Compute the cumulative weekly total in (c) directly, using a single array formula without computing the intermediate numbers of components you need. Display your result as a 1x20 range.

2.3 [25] 

El Capitan Insurance is planning to roll out a new series of annuities this winter. The products are somewhat technical, and there are 5 of them. The Wholesaler Support Group, which has 5 staff, is expected to contact each of El Capitan's 50 wholesalers by telephone during the week of the product introduction to explain the new products to them, and to guide them in choosing target markets.

Two of the Wholesaler Support Group staff, Vincent and Victoria, are experienced veterans. Because they have established personal relationships with the wholesalers, they're more effective and productive than the three rookies on the staff, Rose, Richard and Rachel. Each staffer has 10 wholesaler accounts, but not all wholesalers will sell all products equally. The expected number of hours required by each staffer to brief a wholesaler on each of the five products is as shown in Table 2.3.1.

Note: This problem is designed to give you an idea of how it feels when you spend a lot of time working something out, and then your customer says something like "Remember I told you we had 6 products? Well, we're canceling Number 2. Have a nice day." This is typical. Normally, the rework is major, expensive and, because it's often done in a hurry, it's just as often wrong. I'm trying to show you in this problem how to anticipate the need for flexibility when you build models. That way, you can avoid getting caught in the position of having to make complex major changes close to the deadline because somebody else changed something.

Table 2.3.1 Expected effort required of veterans and rookies to brief a wholesaler on each of the five new products.

Vets Rookies Product A 1.00 1.23 Product B 0.50 0.73 Product C 0.76 1.23 Product D 0.50 0.75 Product E 0.73 1.00

(a) [5] 

The hours of operation of El Capitan Insurance are 8 am to 5 pm Monday through Friday. Everyone on the staff takes an hour for lunch starting at Noon. If all staffers start contacting their wholesalers at 8 am Monday morning, and work 8-hour days, and if they talk to each wholesaler about each product, when will they each finish? State your answers as a day of the week and a time of day for Veterans and for Rookies.

Note: Don't get involved with using Excel's date and time functions, unless you already know how they work. Just compute the hours required for Vets and for Rookies, and then convert that to a day and time using whatever means you can, including pencil and paper. Enter each day and time as text into a cell of the worksheet, one for Vets and one for Rookies.

(b) [10] 

The expected unit sales of Victoria's ten wholesalers for each of the five products are as shown in Table 2.3.2. Assume also (though this is a bit unrealistic) that Rose's ten wholesalers are expected to perform identically.

Table 2.3.2: Expected Unit Sales of new products by Victoria's 10 wholesalers

Note:Before you start this one, read part (d), especially the note.

The reason for posing these two parts in this way is to emphasize that in real life, we often are asked to make "just a small change" to some work we have already done. Although the change might be small, the amount of work required to implement it can be just as great as the work we have done so far (or greater!). If you live in that kind of environment (and most of us do), it helps to set things up so they can be easily changed.

As the manager of the Wholesaler Support Group of El Cap, you decide that it's best for each of your people to be selective in briefing the wholesalers. If Rose and Victoria brief only those expected to sell more than 550 units of a product, and if they brief their wholesalers on only those products, how long will it take each of them to brief each wholesaler (in hours)? Express your result as a 2x10 range, with the top row representing the hours Victoria spends with each of the 10 wholesalers, and the bottom row representing the hours Rose spends with each of the 10 wholesalers.

(c) [5] 

Repeat part (b) for a threshold of 650 units.

(d) [5] 

Marketing is considering dropping Product E. Repeat Part (b) without Product E.

Note: if you were clever in setting up your answer to (b), this one isn't much work.