Math 13 Chapter 3 Handout Helene Payne. Name: 1. Assign the value to the variables so that a matrix equality results.

Transcription

1 Matrices Name:. Assign the value to the variables so that a matrix equality results. [ [ t = x 3. Are the following matrices equal, why or why not? [ 3 7, Let the matrix A be defined by [ 7 3, find 3A. 4. Let the matrices and [ 7 3 [ 9 4 B = 3 C = [ 5.,, (a) Find A B. (b) Find (B A). (c) Find (A + C) Solve the following matrix equation for X. [ [ 3 + X = 3 4 5

2 Inventory Rule Let matrices be defined as follows: B is the beginning inventory, P is the purchases, S is the sales, and E is the ending inventory over a given time interval. Then E = B + P S 6. During a given week, a small convenience store has a beginning inventory given by the following matrix: % Whole Quarts gallon [ 3 8 The sales inventory of the store for the same week is given by the following matrix: = B Quarts gallon % Whole [ 8 5 = S If the ending matrix is B, what is the purchase matrix? 7. Calculate the following row-column product [ Let the matrices A and B be defined by [ 3 and B = (a) Find the matrix product AB [ 3. (b) Find the matrix product BA. (c) Find the matrix product A. Page

3 9. Using matrices from the previous exercise, find the matrices (a) AI (b) IA (c) BI. Let the matrices A, B and C be defined by [ [ [ 4, B =, C = 3 [ 4, and D =. (a) Find AB and BA. (b) Find AC and AD.. The amount of fat (F ), sodium (S), and protein (P ), in units of grams, milligrams and ounces respectively, per serving of a soup, a meat, and a vegetable are a follows: Soup Meat Vegetable F S P = A (a) Given that a person eats serving of soup,.5 servings of meat, and.73 servings of the vegetable, find a matrix B that, when multiplied with A, will give the total intake of fat, sodium, and protein. Then do the matrix multiplication. (b) A person on a strict diet for health reasons gets 3 points for each gram of fat,.5 points for each milligram of sodium, and points for each ounce of protein. Find a matrix C that, when multiplied with A, will give the total points accumulated for each food. Perform the multiplication. Page 3

4 . Use the laws of matrix algebra to factor out any common quantities for each expression. (a) (AB)C + BC (b) AX 3AY 3. (a) Write the following matrix equation as a system of equations. [ 3 5 x = 5 y 4 6 (b) Write the following system of equations as a matrix multiplication. x + z = 4w y + 4z = 4 w + 4x 6y = 4. Decide whether matrix B is the inverse of matrix A, by multiplying the matrices. (a) [ 5 [ 5, and B = (b), and B = 5. We would like to find out if matrix A defined by [ 3 4 has an inverse. We will assume that such an inverse exists and let [ a b A =. c d If we multiply AA we obtain [ [ 3 a b AA = 4 c d [ 3a + c 3b + d = 4a + c 4b + d = [ Page 4

5 Write down the two system of equations resulting from the above multiplication, one for a and c, and one for b and d. Then find the corresponding augmented matrices. The lefthandsides are the same in the two augmented matrices in the above exercise. If each system is solved by the pivoting process using Gauss-Jordan elimination we obtain: [ 3 4 [ a c and [ 3 4 [ b d Since both systems have the same coefficient matrix, write them together in one augmented matrix with two righthandsides. Then solve the two systems by pivoting the extra wide augmented matrix. go from here [ 3 4 to here [ a b c d 6. Verify that the matrix obtained in the previous exercise is the inverse by multiplying it with the original matrix. Finding the Inverse of a Matrix Let A be any n n matrix. To find the inverse if there is one, or to find out there is no inverse, follow the steps below.. Form the n n doublewide matrix, [A I, where I is the identity matrix.. Perform pivot operations on elements on the A side on the whole rows, until your obtain [R B, where R is the reduced row-echelon form. 3. If R = I then B = A. If R I, it will contain a row of zeros and A has no inverse. 7. Find the inverse of the matrix below by forming the double-wide matrix and performing pivoting. 3 Inverse of a Matrix [ a b For the matrix, let D = ad bc. If D =, then A c d does not exist. If D, the A exists and is given by: [ d b A = D c a Page 5

6 8. (a) Find the inverse of the matrix below by using the formula for matrices. [ 3 (b) Solve the following system of equations by writing it as a matrix product and multiplying from the left by the inverse of the coefficient matrix. 3x + y = 8 x + y = 5 Finding the Inverse of a Matrix Using the TI83 Plus Calculator Enter the matrix by pressing nd x, then go to EDIT to enter your matrix, first entering the matrix dimension. When you are done, press nd MODE to exit the matrix edit mode. Go back to the matrix menu by once again pressing nd x, and move up or down the list to select the matrix just entered, then press ENTER. Press the x button, then press enter. If the matrix has an inverse, it will be displayed. If it doesn t have an inverse, you will get an error message. 9. Find the inverse of the matrix below using your calculator A particular police department employs two grades of police personnel: rookies and sergeants. A person at the grade of rookie is to spend hours in training and hours in patrol each week. A person at the grade of sergeant is to spend 5 hours in training and 3 hours on patrol duty each week. The training center can effectively handle 4 person-hours each week, while the department needs at least 44 person-hours each week for patrol duty for the two grades of personnel. How many persons at each of the two grades does the department have, assuming that the training center operates at full capacity and that the minimum requirements for patrol duty are met. (a) Set up a system of equations that model the given problem. (b) Rewrite the systems of equations as the matrix equation AX = B. (c) Use the inverse of matrix A to solve the system of equations. Page 6