Inverse of a Matrix: This activity will illustrate how Excel can help you find and use the inverse of a matrix. It will also discuss a matrix function called the determinant and a method called Cramer's Rule that uses determinants to solve systems of equations. Example 1: (part 1 - do not type into Excel, just read through this example) Given the following matrix A, find A -1. A= 1-2 3 2-1 3 In this example you will be shown the results of finding the inverse of the matrix by hand using row reduction. You have learned (or will soon learn) in the class lectures how to do this. Since A is a square 3x3 matrix, it is possible that A -1 exists. First create a 3x6 matrix composed of matrix A on the left augmented with the I 3 on the right. This is illustrated by matrix C below. C= 1-2 3 1 0 0 2-1 3 0 0 1 Row-reduce the augmented matrix, making sure that the final matrix is in reduced row echelon form. If the left side half of the matrix looks like the 3x3 identity matrix, the right side of the matrix will contain A -1. If the left side of the matrix has either a row or a column consisting of all zeros then the inverse does not exist. Below is the matrix C in reduced row echelon form. Can you find the inverse of A? To test your guess, multiply the matrix A and its inverse to see if you get the identity matrix I. 1 0 0-1 -1 1 0 0 1 2/3 1-1/3 Be sure you understand the above process for finding the inverse matrix. Now you will learn the Excel command to give you the inverse quickly. Example 1: (part 2 work this and following examples in Excel as you read through this problem) Enter matrix A into your spreadsheet. Since A is a 3x3 matrix, its inverse will also be a 3x3 matrix. Choose the position where you want the inverse to appear. In a cell immediately to the left of this position, enter the label "A^(-1)=". Highlight the 3x3 block of cells that will contain the inverse and type the following command: =minverse(b1:d3). Next press the Shift-Ctrl-Enter keys(or Apple-Enter) simultaneously. Your results should resemble the following. A^(-1)= -1-1 1 0.666 1-0.333 Page 1 of 5
Now we will look at an example using the inverse to solve a system of equations in the form AX=B. Example 2: Consider the system: X - 2Y + 3Z = 4 Y = -3 2X - Y + 3Z = 1 Notice that the coefficient matrix A is the same matrix that we used above so we have already found the inverse. Now you only need to enter the 3x1 matrix B into your spreadsheet and perform matrix multiplication to find the solution: X = (A -1 )*B. The commands used are: =minverse(b1:d3) (to find the inverse) and =mmult(b5:d7, B9:B11) (to perform the multiplication). A= 1-2 3 2-1 3 A -1 = -1-1 1 0.666 1-0.333 B= 4-3 1 X= 0-3 -0.666 NOTE: In your final answers you often will have unwieldy decimal answers. You may wish to reduce the number of decimal places. Simply go to the decimal icons and click on the icon until you have the decimal representation you want. You may also highlight the cells and go to Format-Cells-Number and select the desired number of decimal places. Often you will have a value that looks something like 374E-16. This is scientific notation for the value.0000000000000000374 which is essentially the zero value. You may wish to replace this value with zero BUT you must be careful. First you must convert your commands to actual numerical values. As in the last activity, highlight the entire matrix and go to Edit and then Copy. With the same block of cells still highlighted, choose Edit and Paste special. A dialogue box will appear. You should click on values and then OK. This converts all cells to regular numerical values and allows you to delete a value and replace it with another value such as zero. Page 2 of 5
Problems to be handed in: (Using Excel s inverse command to solve systems of equations) 1. Consider the system of equations.25x 1.25x 2 +.5x 3 +.75x 4 = 3.5x 1 +.5x 2 +.5x 4 = 2 x 1 +..25x 2.25x 3.25x 4 = 2.25x 1 +.5x 2 +.75x 3 = 1.5 Use Excel to perform the following operations. Identify your matrices by labeling as in the examples shown in the introduction. (a) Define the coefficient matrix (not augmented) to be matrix A. (b) Define the column matrix of constants to be matrix B. (c) Find A -1. (d) Solve the system using A -1 by multiplying A -1 B. (e) Clearly state the solution to the system. 2. An electronics company produces transistors, resistors, and computer chips. Each transistor requires 3 units of copper, 1 unit of zinc, and 2 units of glass. Each resistor requires 3, 2, and 1 units of the three materials, respectively; and each computer chip requires 2, 1, and 2 units of these materials, respectively. This information may be easier to read by putting it into a table. transistors resistors comp.chips copper zinc glass 3 1 2 3 2 1 2 1 2 The supply of these materials varies from week to week, so the company needs to determine a different production run each week, based on the supplies available. Suppose that for the week beginning October 11 the total amounts of materials available are 810 units of copper, 410 units of zinc, and 490 units of glass. A system of equations is set up to model the production run: 3x 1 + 3x 2 = 810 1x 1 + 2x 2 + 1x 3 = 410 2x 1 + 1x 2 = 490 Note that the equations in this system do not represent the rows of the table, but the columns. When information is in a table, it may or may not be used in exactly that position. Because the amount of copper available is 810 units, the total copper used must add up to that amount, which requires looking at the copper column. Page 3 of 5
To solve the system and find the production run for the first week, the method of solution will use the inverse of the coefficient matrix. This will also allow future production runs to be quickly determined. (a) The symbol definition for the system is given above. Tell what the variables x 1, x 2, and x 3 represent and give the verbal definition for the first equation of the system. That is, describe in words the exact information that the equation is relating to you. (b) Solve the system by following the same steps as a) - e) in problem 1). Write the solution in the form of a brief memo to the production line supervisor from the department manager (you). A few short sentences will suffice, typed beside your work in Excel. (c) For the week beginning October 18 you again have to decide the production run. This time the total amounts available are 700 units of copper, 380 units of zinc, and 450 units of glass. On scratch paper, write the system that must be solved and look at similarities with the first system. Solve, using the inverse. Whenever possible, use your work from previous steps. Do not duplicate your previous work. Write another memo for the new information. 3. Use Excel to solve problem #38, page 270 in your text. 4. Use Excel to solve problem #57, page 270-271 in your text. Determinants and Cramer s Rule: Another matrix concept that is very important is the determinant. This is a tool that can come in very handy. In this example will discuss two uses for it. Example 3: 3 3 2 Does the Matrix C = 1 2 1 2 1 2 have an inverse? If the matrix is a square matrix, it may have an inverse. You can use the Excel s determinant command to determine if an inverse exists. If the value of the determinant is zero then the answer is NO, there is no inverse for the matrix. If the value the determinant is not zero, then the answer is YES and the inverse does exist. In order to use Excel to find the determinant, type the matrix into your spreadsheet. Highlight one empty cell and type in the command =mdeterm(b1:d3) (assuming that your matrix C was entered into the cell block defined by B1:D3) and hit Shift-Ctrl-Enter ( or Apple-Enter). The actual determinant for C is given by det(c) = 3. Verify this value yourself with Excel. Determinants can be used to solve a system of equations if the coefficient matrix of the system has an inverse. Page 4 of 5
Example 4: The following system is the system of equations from problem 2 in the Problems to be handed in set above: 3x 1 + 3x 2 = 810 1x 1 + 2x 2 + 1x 3 = 410 2x 1 + 1x 2 = 490 Below are several determinants associated with this system. Can you see the pattern used to create the determinants? Notice the determinant array is always written with straight sides instead of the square brackets used for matrices. A = 3 3 2 1 2 1 2 1 2 C = 810 3 2 410 2 1 490 1 2 D = 3 810 2 1 410 1 2 490 2 E = 3 3 810 1 2 410 2 1 490 In order to solve the system you will set up your unknowns as follows: x 1 = C/A x 2 = D/A x 3 = E/A Use Excel to find the values of each of the determinants A, C, D, E. Plug in these values into the appropriate equations from above to find x 1, x 2 and x 3. Did you get the same solution for this system as before? This method of solving a system of equations is called Cramer s Rule. Problems to be handed in: (continued) 5. Use Excel to complete problem #47 on page 270 in your text. 6. Use Excel and Cramer's Rule to redo Example 4) using the value changes given in problem #2, part for the constants: 700, 380 & 450. Page 5 of 5