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1 Question 1: How is information organized in a matrix? When we use a matrix to organize data, all of the definitions we developed in Chapter 2 still apply. You ll recall that a matrix can have any number of rows and columns and is typically named with a capital letter. A matrix with m rows and n columns named A would look like a a a a a a a a A a a a a n n m1 m2 m3 mn Notice that this matrix does not contain a dashed vertical line in front of the last column. he dashed line is unique to augmented matrices and is used to separate the coefficients from the constants. he dots in the matrix indicated a pattern in the matrix. In this case, the dots indicate the arbitrary number of rows m and columns n in the matrix. he individual entries (also called elements) of the matrix are symbolized with lowercase letters, like a mn, and these symbols represent numbers. he subscript on the lowercase letter indicates the location of the entry in the matrix. he symbol a 23 represents the number in the second row, third column of the matrix. he size of a matrix (also called the dimensions of the matrix) is the number of rows and columns in a matrix. For the matrix A with m rows and n columns, we would say the size of the matrix is m x n (read m by n). Several sizes of matrices are given special names. A matrix with the same number of rows and column is called a square matrix. An example of a square matrix is the 2 x 2 matrix 2

2 he exact number of rows and columns in a square matrix is not important, only the fact that the number of rows and columns is the same. Matrices with a single row or a single column are also given special names. Row matrices like or are matrices with only a single row, but any number of columns. Column matrices like 2 3 or are matrices with any number 4 10 of rows, but a single column. he size of a matrix is an important prerequisite in determining if two matrices are equal. wo matrices are equal if they have the same size and each entry in one matrix is equal to the corresponding entry in the other matrix. Example 1 Matrix erminology he matrices are 3 x 3 square matrices A B a. What is the value of the entry b 32? 3

3 Solution he subscript on b 32 refers to the entry in the third row, second column of the matrix B. herefore, b b. Is a23 b23? Solution he subscripts on a 23 and b 23 refer to the corresponding entries in the second row, third column of A and B. Since the entry in that location is 4 in both matrices, a23 b23 and is equal to 4. c. Is A B? Solution For the matrices to be equal, they must have the same size and each entry in A must be equal to the corresponding entry in B. Both matrices are 3 x 3. In part b, we determined that the entries in the second row, third column of each matrix were equal. However, a 13 in A and b 13 in B are not the same so A B. his is in spite of the fact that every other set of corresponding entries are equal. Ed Magazine is a fictional magazine that publishes four issues each year. It has a loyal base of subscribers and twice a year it conducts subscription drives for new subscribers. At the same time they are acquiring new subscribers, the current 4

4 subscriber s subscriptions are expiring. Some of these expiring subscriptions belong to first time subscribers and others are long time subscribers who have renewed their subscriptions in the past. he table below shows the numbers of new and expiring subscribers by quarter. New s Expiring s First ime Long ime Quarter Ending 3/ Quarter Ending 6/ Quarter Ending 9/ Quarter Ending 12/ Although this information could be placed in a matrix in several different ways, two approaches stand out. Since the rows in the table correspond to the four different quarters during the year and the columns correspond to numbers of subscribers, we could use a matrix with four rows and three columns: New First Long Q1 Q2 Q3 Q Normally we don t include the red labels on a matrix. However, they are often included to help clarify how the information in the matrix is organized. o name this matrix of subscribers, we could use the letter S and write S

5 his organization capitalizes on the fact that all of the numbers in the table indicate the number of subscribers in a certain category. Let s look at the table differently. New s Expiring s First ime Long ime Quarter Ending 3/ Quarter Ending 6/ Quarter Ending 9/ Quarter Ending 12/ he rows in the table still refer to quarters, but now the shading in the table emphasizes a difference in the numbers. he numbers in the blue region corresponds to the number of new subscribers by quarter and the red region corresponds to subscribers whose subscriptions are expiring. With this difference in mind, we could define two matrices for this table, N, E he matrix N is a 4 x 1 column matrix representing the number of new subscribers of Ed Magazine. he matrix E is a 4 x 2 matrix representing the number of expiring subscribers in two categories by quarter. Depending on the application, these matrices may be more useful than the 4 x 3 matrix S. 6

6 Example 2 Organize Information in a Matrix A magazine s circulation is the number of issues it distributes. Ed Magazine is distributed to three categories of subscribers each quarter. New Renewing Non-renewing Quarter Ending 3/ Quarter Ending 6/ Quarter Ending 9/ Quarter Ending 12/ Use this information to define three matrices named C 1, C 2, and C 3, where C 1 describes the number of issues distributed to new subscribers, C 2 describes the number of issues distributed to subscribers who have renewed their subscription, and C 3 describes the number of issues distributed to subscribers who have not renewed their subscriptions. Solution he first column in the table corresponds to issues distributed to new subscribers New Quarter Ending 3/ Quarter Ending 6/ Quarter Ending 9/ Quarter Ending 12/

7 If we let the rows in the matrix C 1 correspond to the quarters, then can organize the information in the table in a 4 x 1 matrix as C Alternatively, we could also let the quarters correspond to the columns in a 1 x 4 matrix and define C Either matrix organizes the information appropriately. Since the original table matches each row with a quarter, we ll follow the same principal and let the rows of the matrices correspond to the quarters. Renewing Non-renewing Quarter Ending 3/ Quarter Ending 6/ Quarter Ending 9/ Quarter Ending 12/ Letting the rows match the quarters, the other columns in the table give the entries in C 2 and C 3, C C

8 In Example 2, we mentioned the fact that the data in the first column of the table could be written as a row matrix or a column matrix. hese matrices are examples of transposes. In other words, the matrix is the transpose of the matrix he transpose of the matrix A, written A, is obtained by writing the columns of the matrix A as rows in the matrix A. Alternatively, we could also write the rows of the matrix A as columns in the matrix A. Example 3 Find the ranspose of a Matrix Find and label the transpose of the expiring subscriber matrix E Solution o get the transpose of the subscriber matrix, we interchange the rows and columns. In other words, the columns of E become the rows of the transpose transpose E to yield E or the rows of E become the columns of the E

9 In the original matrix E, the rows of the subscriber matrix correspond to the quarters and the columns tell us the subscriber category. Renewing Non-renewing s Suscribers Spring Quarter Summer Quarter Fall Quarter Winter Quarter In the transpose, these roles are reversed. Spring Summer Fall Winter Quarter Quarter Quarter Quarter Renewing s Non-renewing s he information in each matrix is the same, but organized differently. 10