 Hello and welcome to the session. In this session we are going to discuss the following question which says that Suppose the demand curve for a sewing machine over some time period can be written as x is equal to 3.8 minus 0.4 y Where x is the price of an automobile and y is the corresponding quantity Suppose that the supply curve is x is equal to 4.75 minus 0.375 y Use matrix theory to obtain x and y We are given the demand curve for a sewing machine over some time period which is given by x is equal to 4.8 minus 0.4 y And the supply curve is given by x is equal to 4.75 minus 0.375 Where x is the price of an automobile and y is the corresponding quantity Using matrix theory we need to find out the values of x and y If we are given two simultaneous equations in two variables x and y such that a1x plus b1y is equal to c1 and a2x plus b2y is equal to c2 Now these system of equations can be written in the matrix form as The 2 by 2 matrix containing elements a1 a2 b1 b2 into 2 by 1 column matrix containing elements xy is equal to 2 by 1 column matrix containing elements c1 c2 or can also be written as Ax is equal to b where a is the 2 by 2 matrix can be at the 2 by 1 column matrices With this key idea we shall proceed with the solution We are given the system of equations as x is equal to 4.8 minus 0.4 y and x is equal to 4.75 minus 0.375 The given system of equations can be written as x plus 0.4 y is equal to 4.8 and x plus 0.375 y is equal to 4.75 Now if we write these equations in the matrix form we get The 2 by 2 matrix containing elements 1 1 0.4 0.375 into 2 by 1 column matrix containing elements xy is equal to 2 by 1 column matrix containing elements 4.8 4.75 From the written in the form Ax is equal to b is the 2 by 2 matrix containing elements 1 1 0.4 0.375 is the 2 by 1 column matrix containing elements xy and b is the 2 by 1 column matrix containing elements 4.8 4.75 Now Ax is equal to b implies that xy is equal to A inverse of b Now inverse of matrix A that is inverse of 2 by 2 matrix containing elements 1 1 0.4 0.375 is equal to 1 by determinant of A into a joint of A Now determinant of A is given by 1 into 0.375 that is 0.375 minus of 1 into 0.4 that is 0.4 which is equal to minus of 0.025 so determinant of A is equal to minus of 0.025 and A is given by the 2 by 2 matrix containing elements 1 1 0.4 0.375 and a joint of A can be obtained by interchanging the entries on the leading diagonal and changing the sign of the entries on the other diagonal therefore a joint of A is given by the 2 by 2 matrix containing elements 0.375 minus of 0.4 minus 1 1 Now inverse of matrix A that is inverse of 2 by 2 matrix 1 1 0.4 0.375 is equal to 1 by determinant of A that is minus of 0.025 into inverse of A that is 2 by 2 matrix containing elements 0.375 minus 1 minus 0.4 1 which is equal to 1 upon minus of 0.025 can be written as minus 30 into 2 by 2 matrix containing elements 0.375 minus 1 minus 0.4 1 which is equal to the 2 by 2 matrix containing elements minus 30 into 0.375 that is minus 15 minus 30 into minus 1 that is 30 minus 30 into minus 0.4 that is 16 minus 30 into 1 that is minus 30 therefore inverse of A is given by the 2 by 2 matrix containing elements minus 15 40 16 minus 40 we know that X is given by A inverse into B we have X is equal to A inverse into B where X is the 2 by 1 common matrix containing elements X Y is equal to A inverse that is the 2 by 2 matrix containing elements minus 15 40 16 minus 40 into B B is the 2 by 1 column matrix containing elements 4.8 4.75 which is equal to the 2 by 1 matrix containing elements minus 15 into 4.8 plus 16 into 4.75 40 into 4.8 plus minus 40 into 4.75 which is equal to the 2 by 1 matrix containing elements minus 72 plus 76 192 minus 190 which is equal to the 2 by 1 matrix containing elements first 2 now 2 by 1 matrix containing elements X Y is equal to the 2 by 1 matrix containing elements 4.2 which implies that X is equal to 4 and Y is equal to 2 which is the required answer this completes our session hope you enjoyed this session