Gem in the Matrix

Introduction

In the gem mines of western North Carolina, dynamite is used to break the mineral-rich rock into manageable-sized pieces. Later, the pieces of stone are examined to see which ones contain gems. Eventually, the stonecutter removes the gems from their matrix (the stone that contains the gem) and cuts them into gemstones. The company, Gem Valley Mines (GVM), needs to know which parts of the mine are most productive and profitable, so the manager keeps monthly records of the number of gemstones mined each month and the precise location of the vein in the rock from which they came. His records for the months of January and February are shown in the tables below.

January

Vein / Emeralds / Sapphires / Aquamarines
A / 45 / 61 / 112
B / 27 / 46 / 131
C / 37 / 33 / 78

February

Vein / Emeralds / Sapphires / Aquamarines
A / 52 / 73 / 99
B / 36 / 52 / 103
C / 45 / 39 / 83

Part 1

1.  Fill in the cart below to give the total number of gemstones mined from each vein for the months of January and February

Vein / Emeralds / Sapphires / Aquamarines
A
B
C

Information is often presented in charts consisting of rows and columns of numbers, as the GVM company has done. For publication, it is important to label both rows and columns so that the charts can be interpreted by those who do not normally work with them. However, since the Gem Valley Mines manager and staff know what each entry represents, they only record the actual array of numbers in row-and-column format for their internal work purposes.

For example, the internal working array for the months of January and February, respectively are presented as shown below.

J = 45611122746131373378 and F = 5273993652103453983

In mathematics, an array of numbers in rows and columns, such as the gemstone company uses, is called a matrix. Employees of GVM know that each row tells which vein the gemstones came from and each column identifies the type of gemstone. Note: The plural of matrix is matrices.

2.  Use the two matrices shown above to answer the following questions.

  1. How many emeralds came from vein B in January? _____
  1. How many sapphires came from vein C in February? _____

3.  In question 1, you completed a chart to show the combined results of production for the months of January and February.

  1. Write this combined information as matrix C.

C =

  1. Each number in the matrix is called an entry. Explain how the entries found in matrices J and F were used to determine the entries in matrix C. Your explanation should discus not only the numerical values but also the location of these values in matrix C.

Part 2

4.  The quarterly report for the first three months of the year is represented by matrix Q, as shown below.

Q = 1502133059514834013198213

  1. What does the entry “340” mean in matrix Q?
  1. What does the entry “131” mean in matrix Q?

5.  Matrix C, in Question 3, gives the total number of gemstones mined from each vein for the months of January and February. Matrix C is shown below.

C = 9713421163982348272161

  1. Use matrix C and the quarterly report matrix Q to create a matrix that shows the number of each type of gemstone produced from each vein for the month of March. Call this matrix M.

M =

  1. How many sapphires were mined from vein B during the month of March?

6.  The GVM manager always produces quarterly reports that summarize production in several ways. One aspect of his summary is the reporting of average monthly gemstone production.

  1. Create matrix A to show the average monthly gemstone production for the first quarter of the year.

A =

  1. What is the monthly average for the number of aquamarines that were mined from vein A during the quarter of January, February, and March?
  1. What operations did you perform on the entries in matrix Q to get the entries in matrix A?

In Questions 4 – 6, you discovered that corresponding entries of matrices can be added and subtracted, and each entry can even be multiplied by the same number to obtain a new matrix. For example, to obtain the entries in matrix A, each entry in matrix Q was multiplied by the number 13. When a new matrix is formed by multiplying each entry of an original matrix by a number, called a scalar, the process is called scalar multiplication. Another example to illustrate scalar multiplication is shown below.

2241410536 = 48282010612

Notice that each entry shown in the original matrix above is multiplied by 2 to get the corresponding entry of the resulting matrix.

7.  Matrices A, C, F, J, and Q refer to the matrices defined in the previous questions in this Unit.

  1. Show that J + F + (Q – C) = 3A
  1. Explain why J + F + (Q – C) is equal to 3A in terms of what the matrices represent.

Part 3

Recall that matrix J, shown below, represents the gemstones mined in January from the three veins. Matrix K, also shown below, measures the total carat weight of the emeralds, sapphires, and aquamarines mined from these three veins in January. Carat is a weight measurement and the weight of gemstones is always represented in carats.

J = 45611122746131373378 K = 406098298078256845

8.  Matrices J and K each give information about three kinds of gemstones from three veins. The clerk entered different matrices in the computer and printed out some new matrices. One was matrix J + K, shown below. Is there useful information contained in this matrix? Provide a rationale to support your answer.

J + K = 851212105612620962101123

Suppose the manager has matrix V, shown below, with three rows and two columns, where the rows denote the veins A, B, and C, respectively, and the columns denote gemstones mined in April for two of the three gemstone types. The data for the third gemstone category is not yet available.

V = 378322943056

9.  What difficulties arise if the manager attempts to add matrix V to matrix Q? Explain your reasoning?

10.  Based on your work in answering Questions 8 and 9, under what conditions does it make sense to add two matrices? Explain your reasoning.

Part 4

11.  At the end of the second quarter, the comptroller discovers that she is missing the monthly report for May. The comptroller can find the average for the quarter and the data for April and June. Help her organize the data into matrices to reconstruct the report for May. Explain the process of finding the missing data for the month of May.

April data: Vein A: Emeralds 56, Sapphires 61, Aquamarines 111

Vein B: Emeralds 42, Sapphires 45, Aquamarines 126

Vein C: Emeralds 31, Sapphires 33, Aquamarines 115

June data: Emeralds: Vein A 73, Vein B 33, Vein C 22

Sapphires: Vein A 39, Vein B 42, Vein C 99

Aquamarines: Vein A 86, Vein B 103, Vein C 99

Second Quarter Vein A: Emeralds 61, Sapphires 46, Aquamarines 92

Averages Vein B: Emeralds 31, Sapphires 40, Aquamarines 106

Vein C: Emeralds 28, Sapphires 52, Aquamarines 86