TABLE 1 Gallons of Crude Oil Requiredto Produce1 Gallon of Gasoline Crude Premium Regular Reqular Unleaded Unleaded Leaded 14 Many computations that commonly occur in operations research(and other branches of mathematics)can be concisely expressed by using matrix multiplication.To illustrate this,suppose an oil company manufactures three types of gasoline:premium unleaded, regular unleaded,and regular leaded.These gasolines are produced by mixing two types of crude oil:crude oil 1 and crude oil 2.The number of gallons of crude oil required to manufacture 1 gallon of gasoline is given in Table 1. From this information,we can find the amount of each type of crude oil needed to manufacture a given amount of gasoline.For example,if the company wants to produce 10 gallons of premium unleaded,6 gallons of regular unleaded,and 5 gallons of regular leaded,then the company's crude oil requirements would be Crude1 required=((10)+(f)(⑥)+（分5=12.75 gallons Crude 2 required =((10)+()(6)+()5 =8.25 gallons More generally,we define pu=gallons of premium unleaded produced ru gallons of regular unleaded produced rL=gallons of regular leaded produced cI=gallons of crude 1 required c2 gallons of crude 2 required Then the relationship between these variables may be expressed by c=(经)Pu+()ru+()rL c2=()Pu+(ru+(孕rz Using matrix multiplication,these relationships may be expressed by 3 21 Properties of Matrix Multiplication To close this section,we discuss some important properties of matrix multiplication.In what follows,we assume that all matrix products are defined. 1 Row i of 4B =(row i of A)B.To illustrate this property,let 「117 4-61 and B=23 L12 Then row 2 of the 2 X 2 matrix AB is equal toMany computations that commonly occur in operations research (and other branches of mathematics) can be concisely expressed by using matrix multiplication.To illustrate this, suppose an oil company manufactures three types of gasoline: premium unleaded, regular unleaded, and regular leaded. These gasolines are produced by mixing two types of crude oil: crude oil 1 and crude oil 2. The number of gallons of crude oil required to manufacture 1 gallon of gasoline is given in Table 1. From this information, we can find the amount of each type of crude oil needed to manufacture a given amount of gasoline. For example, if the company wants to produce 10 gallons of premium unleaded, 6 gallons of regular unleaded, and 5 gallons of regular leaded, then the company’s crude oil requirements would be Crude 1 required ( 3 4 ) (10)  ( 2 3 ) (6)  ( 1 4 ) 5 12.75 gallons Crude 2 required ( 1 4 ) (10)  ( 1 3 ) (6)  ( 3 4 ) 5 8.25 gallons More generally, we define pU gallons of premium unleaded produced rU gallons of regular unleaded produced rL gallons of regular leaded produced c1 gallons of crude 1 required c2 gallons of crude 2 required Then the relationship between these variables may be expressed by c1 ( 3 4 ) pU  ( 2 3 ) rU  ( 1 4 ) rL c2 ( 1 4 ) pU  ( 1 3 ) rU  ( 3 4 ) rL Using matrix multiplication, these relationships may be expressed by Properties of Matrix Multiplication To close this section, we discuss some important properties of matrix multiplication. In what follows, we assume that all matrix products are defined. 1 Row i of AB (row i of A)B. To illustrate this property, let A and B Then row 2 of the 2 2 matrix AB is equal to 1 3 2 1 2 1 2 3 1 1 1 2 pU rU rL  1 4   3 4   2 3   1 3   3 4   1 4  c1 c2 18 CHAPTER 2 Basic Linear Algebra TAB LE 1 Gallons of Crude Oil Required to Produce 1 Gallon of Gasoline Crude Premium Regular Regular Oil Unleaded Unleaded Leaded 1  3 4  2 3  1 4  2  1 4  1 3  3 4