MATH PLUS STUDY SKILLS WORKSHOP



MATH 1324 4.3 4.4 Date: A matrix is said to be in reduced row echelon form, if1. Each row consisting entirely of zeros is below any row having at least one nonzero element.2. The leftmost nonzero element in each row is 1.3. All other elements in the column containing the leftmost 1 of a given row are zeros.4. The leftmost 1 in any row is to the right of the leftmost 1 in the row above. Elementary Row Operations to transform linear systems into equivalent systems:(A) Two equations are interchanged.(B) An equation is multiplied by a nonzero constant.(C) A constant multiple of one equation is added to another equation.Procedure Gauss–Jordan EliminationStep 1: Choose the leftmost nonzero column and use appropriate row operations to get a 1 at the top.Step 2: Use multiples of the row containing the 1 from step 1 to get zeros in all remaining places in the column containing this 1.Step 3: Repeat step 1 with the submatrix formed by (mentally) deleting the row used in step 2 and all rows above this row.Step 4: Repeat step 2 with the entire matrix, including the rows deleted mentally. Continue this process until the entire matrix is in reduced form.Solve by Gauss–Jordan elimination: 2x – 2y + z = 3; 3x + y - z = 7; x – 3y + 2z = 0Write the augmented matrix and follow the stepsSteps to solve using calculator: 1. Go to Matrix > edit > enter order, elements > exit 2. Go to Matrix > math > rref > matrix > select> enter 3. Find your solution on the calculator.Solve by Gauss–Jordan elimination: 3x1 + x2 - 2x3 = 2; x1 - 2x2 + x3 = 3; 2x1 - x2 - 3x3 = 3Solve by Gauss–Jordan elimination: (Use calculator)2x1 - 4x2 + x3 = -4; 4x1 - 8x2 + 7x3 = 2; -2x1 + 4x2 - 3x3 = 53x1 + 6x2 - 9x3 = 15; 2x1 + 4x2 - 6x3 = 10; -2x1 - 3x2 + 4x3 = -6A company that rents small moving trucks wants to purchase 25 trucks with a combined capacity of 28,000 cubic feet. Three different types of trucks are available: a 10-foot truck with a capacity of 350 cubic feet, a 14-foot truck with a capacity of 700 cubic feet, and a 24-foot truck with a capacity of 1,400 cubic feet. (A) How many of each type of truck should the company purchase?(B) The rental company charges $19.95 per day for a 10-foot truck, $29.95 per day for a 14-foot truck, and $39.95 per day for a 24-foot truck. Which of the four possible choices in the table would produce the largest daily income from truck rentals?4.4 Addition: If the matrices A and B have the same size, then their sum is the matrix A+B defined by (A + B)ij = aij + bij.1-2423-36-44 + -543 012-561 = Their difference is the matrix A ? B defined by (A ? B)ij = aij? bij1-2423-36-44 - -543012-561 = Difference: A matrix A can be multiplied by a scalar c to obtain the matrix cA, where (cA)ij= caij. This is called scalar multiplication. We just multiply each entry of A by c. For example51-2423-36-44 - 2 -543012-561 = - = Sales Commissions Ms. Smith and Mr. Jones are salespeople in a new-car agency that sells only two models. August was the last month for this year’s models, and next year’s models were introduced in September. Gross dollar sales for each month are given in the following matrices:What were the combined dollar sales in August and September for each salesperson and each model?What was the increase in dollar sales from August to September?If both salespeople receive 5% commissions on gross dollar sales, compute the commission for each person for each model sold in September.Dot product of two vectors (row-column): Multiply the corresponding elements of rows of first matrix and columns of second matrix then sum all the terms. For this number of columns in the first need to be equal to the number of rows of second matrix. u= 120-12 v= 2 110-5 4Product of two matrices: If the number of columns of A equals the number of rows of B, then the product ABis defined byABi j= ai1b1j+ ai2b2j+ ai3b3j + …..aikbkjHere k is the number of columns of A or rows of B.Find the product 123-104-104213 = Find the product AB and BA if possible. What are the order of AB and BA?A= 123041 and B= 104-3-1320Inventory value. A personal computer retail company sells five different computer models through three stores. The inventory of each model on hand in each store is summarized in matrix M. Wholesale (W) and retail (R) values of each model computer are summarized in matrix N.What is the retail value of the inventory at store 2?What is the wholesale value of the inventory at store 3?Discuss possible interpretations of the elements in the matrix products MN and NM.If either product MN or NM has a meaningful interpretation, find the product and label its rows and columns. Discuss methods of matrix multiplication that can be used to find the total inventory of each model on hand at all three stores. State the matrices that can be used and perform the necessary operations.Discuss methods of matrix multiplication that can be used to find the total inventory of all five models at each store. State the matrices that can be used and perform the necessary operations. ................
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