Math 1526: Excel Lab 5 Summer 2004

Transcription

1 Math 5: Excel Lab 5 Summer Matrix Algebra In order to complete matrix operations using Excel, you must learn how to use arrays. First you must learn how to enter a matrix into a spreadsheet. For practice, use the following matrix: A = When you enter matrices into Excel in order to perform matrix operations, it is important that you label the matrices so that they can be easily identified. Matrices can be entered in the spreadsheet anywhere that you choose, but be sure to leave room for the label of the matrix. To enter matrix A, shown above, begin by typing the label A= into the cell of your choice. Next type the matrix entries into the x 5 block of cells that lie immediately to the right of the label. Enter the numbers, one number per cell, in the same order that they appear in the matrix. When you finish, your work should look like the following: Now type in matrix B: (We put ours directly below matrix A in our spreadsheet.) B = Matrix Addition in Excel: You now have two x 5 matrices in your spreadsheet. Since A and B have the same dimension, they can be added together and the resulting matrix, A + B, will also be a x 5 matrix. To have Excel add the two matrices, first choose the location where you want your answer (and its label) to appear. Type in the label A + B =. Immediately to the right of the label, highlight the block of cells that will contain the answer. For this example, this will be a x 5 block of cells.

2 While the block of cells is still highlighted, type the following command: = B:F + B5:F7. But do not press Enter yet! Now hold down Shift and Ctrl keys and press Enter (the keys are held down at the same time). This key combination will calculate all entries for the matrix at the same time. Pressing the Enter key alone (instead of the three-key combination) will only calculate the first entry of the matrix. Comments:. B:F represents matrix A by giving the upper left corner of the matrix (cell B - containing the a entry) and the lower right corner of the matrix (cell F - containing the a 5 entry). B5:F7 represents matrix B in the same manner. Make sure that your values match the location of your matrix.. The Shift-Ctrl-Enter key combination works for PC's. If you are using a Mac computer, you can use Shift- Ctrl-Return or Apple-Enter (NOT Apple-Return). On Macintosh computers, the Return key and the Enter key are not the same thing!

3 IMPORTANT NOTES:. After you have created a new matrix using a matrix command such as addition, you may find that you need to make a change in one of the matrix entries. You cannot do this until the entries have been transformed into "values". To transform the entries into "values", highlight the entire matrix and then select Edit-Copy followed by Edit-Paste-Special-Values. (You are pasting the "values" over your original matrix). You now have a matrix with entries that you can edit.. If you forget and try to make a change to a matrix entry that has not been transformed to "values", the process often becomes "hung-up". There are a couple of options that you might try. First, click OK on the warning that appears on your screen and then try the escape key (Esc). Another possibility that you might try is to use UNDO (from either an icon or from the Edit menu) along with Apple-Enter or Esc. Sometimes, however, nothing seems to help and you are still stuck. In that case, save your program, quit Excel, and then start the Excel program again.. Sometimes you want to copy an existing matrix that was created using a matrix command because you want to edit this matrix and use somewhere else in your work. In this case, when pasting the matrix that you copied, don't forget to use Edit-Paste-Special-Values. Otherwise, you will have problems similar to those described in item above. Scalar Multiplication using Excel: The next example shows you how to use scalar multiplication or multiplying a matrix by a constant. Using matrix A from above, you will use Excel to calculate A. Since matrix A is a x 5 matrix then A is also a x 5 matrix. As you did previously, highlight the correct number of cells needed for your answer. Next type the command = *(B:F) and hit Shift-Ctrl-Enter ( or Apple-Enter on a Mac). You should get the following results: Matrix Multiplication using Excel: Excel can also be used to multiply matrices. Define matrices C and D in Excel as follows: C = 5 and D = 8 Matrix C is a x matrix and D is a x matrix. Since the number of columns in matrix C equals the number of rows in matrix D, it is possible to perform the multiplication CD. The resulting matrix will have the same number

4 of rows as matrix C and the same number of columns as matrix D so the dimension of the resulting matrix will be x. As before, type in your label for your new matrix and highlight a x block of cells where you want your answer to appear. Once you have highlighted the cells, type in the command =mmult(b:g,b5:c) and hit Shift-Ctrl- Return (or Apple-Enter on a Mac). (B:G defines matrix C and B5:C defines matrix D). You should get the following results: Transpose of a Matrix using Excel: The transpose of a matrix A, denoted by A T, is simply a new matrix in which the rows in matrix A become the columns in A T and the columns in matrix A become the rows in A T. Consider the following 5 x matrix A: 8 5 A = 9 5 Since A is a 5 x matrix, A T will be a x 5 matrix. Type in your label for A T and highlight the x 5 block of cells that you want to contain your answer. Type in the command =transpose(b:d5) and hit Shift-Ctrl-Enter (or Apple-Enter). Check your answer against the following:

5 Matrix Application Problems: Suppose you have the matrix multiplication AB = C. If A is an m x n matrix and B is an n x p matrix, then it is possible to multiply matrix A times matrix B and get the resulting matrix C because the number of columns in matrix A equals the number of rows in matrix B. When working with matrices in application problems, it is not enough to be sure that it is physically possible to complete the multiplication. In application matrices, each row and column have descriptive headings. For a matrix multiplication to make sense, you must also have correspondence in these headings. In the following example, you will note that matrix M is a (food) x (restaurant) matrix in that the row headings are foods and the column headings are restaurants. Matrix N is a (food) x (club) matrix. While each matrix is a x matrix and it would be physically possible to obtain the product MN, the values in the resulting matrix have no meaning. For a matrix multiplication AB = C to give meaningful information, it must be true that the column headings in matrix A must match the row headings in matrix B. Check out the following example to see how this is dealt with. Example: The prices (in dollars) of pizza, soda, and salad at three local pizza parlors are summarized in the first table and the orders from three campus groups are summarized in the second table. Let the numerical values in the first table make up matrix M (food x restaurant) and the numerical values in the second table make up matrix N (food x club). Decide how to multiply these two matrices together to find the amount each group would be charged for its entire order at each pizza parlor. Then decide where each group should place its order. Note: One of the matrices will have to be transposed before multiplying in order to get the correct information. Price Orders Papa Johns Pizza Hut VP Pie Math Club Student Gov. Rugby Team Pizza Pizza 5 Soda...5 Soda Salad.95.. Salad Type your matrices into Excel. You should make note of the fact that it is very important that you label the rows and columns of your matrices in Excel when working with application problems so that the work can easily be interpreted either by you in the future or by others who are reviewing your work. Papa Johns Pizza Hut VP Pie M = Pizza Soda...5 Salad.95.. Math Club Student Gov. Rugby Team N = Pizza 5 Soda Salad Since you want to know how much it would cost each club to eat at each of the restaurants, multiplying the price matrix times the order matrix seems to be the correct operation, but this operation will not give useful information since the column headings of matrix M do not match the row headings of matrix N. It is necessary to transpose matrix M so that both the column headings for M T (the matrix on the left) and the row headings for N (the matrix on the right) are the same. With the transformation, you are multiplying (M T )N or a (restaurant) x (food) matrix times a (food) x (club) matrix. The resulting matrix is a (restaurant) x (club) matrix. Note that the final matrix takes its row headings from M T and its column headings from N. 5

6 Pizza Soda Salad M^T = Papa Johns Pizza Hut 9... VP Pie Math Club Student Gov. Rugby Team (M^T)N = Papa Johns Pizza Hut VP Pie It should be easy for you to see from the final matrix that it is cheaper for all three clubs to eat at Papa Johns. Problems to turn in: Work through the example problems in the lab introduction before attempting these problems. Do not turn in the worked examples from the introduction. Also, do not hand in this document with your lab. Work all problems in order. Do not say see attached or other notation referring the grader to another location in the lab for part of a problem. Label problems and parts of problems with appropriate numbers and/or letters. Everything on this lab should be typed. No hand-written work allowed. Use Excel and Excel's matrix operations to complete all of the following problems.. Let A = and B = (a) Use Excel to find A - B. (Remember this is also A+ (-)*B). (b) Use Excel to find the product BA T. (Note: First you will have to find A T. B is x and A T is x, so BA T will be x.) For the following exercises label all your matrices so that it is clear what each matrix represents. That is, if you are multiplying matrix A times matrix B, your label should be AB =. Also label rows and columns appropriately within the context of the problem. Give your answers in complete sentences typed on the lab sheet.. Burger Barn's three locations sell hamburgers, fries, and soft drinks. Barn I sells 9 burgers, orders of fries, and 75 soft drinks each day. Barn II sells 5 burgers a day and Barn III sells 5. Soft drink sales number 9 a day at Barn II and 85 a day at Barn III. Barn II sells 95 and Barn III sells 8 orders of fries per day. (a) Set up a x matrix S that displays daily sales figures for all locations. (b) Burgers sell for $.5 each, fries for $.9 an order, and soft drinks are $. each. Write a x matrix P that displays the prices. (c) Using your matrices from (a) and (b), what matrix multiplication will calculate the daily revenue at each of the three locations? Perform this matrix multiplication to determine the daily revenue. Using the matrix that results from your multiplication, what is the total daily revenue from all locations? (Answer in a complete sentence.)

7 . A furniture manufacturer produces four styles of chairs: style I, style II, style III, and style IV. The following table lists the number of units of raw materials the manufacturer needs for each style. Wood Nylon Velvet Springs Style I 5 Style II 5 Style III 5 Style IV 5 (a) Create matrix A to correspond to the above table. (b) The furniture manufacturer has orders for chairs of style I, of style II, of style III, and of style IV. Create a x matrix B containing the number of units of each style ordered. (c) Use matrix multiplication BA to determine the total number of units of each raw material needed. State your answer in a complete sentence. (d) Suppose wood costs $5 per unit, nylon costs $ per unit, velvet costs $ per unit, and springs cost $ per unit. Create a x matrix C representing the costs for the raw materials. (e) Use matrix multiplication AC to determine the total cost of raw materials for each style of chair. State your answer in a complete sentence. (f) Use matrix multiplication B(AC) to find the x matrix that represents the total investment to fill the orders. State your answer in a complete sentence. 7