 Hello and welcome to the session. In this session we are going to discuss the following question which says that Our company operates best services between two cities. Data regarding passenger traffic during three days is as follows On the first day there were 3 senior citizens, 5 males and 12 females and total revenue is $420 On second day there were 10 senior citizens, 6 males, 4 females and total revenue is $360 On third day there were 9 senior citizens, 7 males and 6 females and total revenue is $420 Find the fare charged for each senior citizen, male and female using matrix theory If we are given 3 simultaneous equations in 3 variables x, y and z that is a1x plus b1y plus c1z is equal to d1 a2x plus b2y plus c2z is equal to b2 a3x plus b3y plus c3z is equal to b3 Then these system of equations can be written in the matrix form as ax is equal to b where a is equal to the 3 by 3 matrix containing elements a1, a2, a3, b1, b2, b3, c1, c2, c3 x is the 3 by 1 matrix containing elements x, y, z and b is the 3 by 1 matrix containing elements d1, d2, d3 x is equal to b implies that x is equal to a inverse into b With this key idea we shall proceed with the solution Let us denote the fare charged by the senior citizens bf by the males bm by the females bf Then we are given the following system of equations We are given in the question that a company operates the services between 2 cities On the first day there were 3 senior citizens 5 males and 12 females and their total revenue is $420 So we have 3s plus 5n plus 12f is equal to $420 On the second day there were 10 senior citizens 6 males 4 females and their total revenue is $360 That is 10s plus 6m plus 4f is equal to 360 On third day there were 9 senior citizens 7 males and 6 females and their total revenue is $420 That is 9s plus 7m plus 6f is equal to 420 And we need to find the fare charged for each senior citizen male and female using matrix theory And from the key idea we know that if we are given 3 simultaneous equations in 3 variables x, y and z The equations are of the form a1x plus b1y plus c1z is equal to d1 a2x plus d2y plus c2z is equal to d2 a3x plus b3y plus c3z is equal to d3 Then these system of equations can be written in the matrix form as ax is equal to b Where a is given by the 3 by 3 matrix containing elements a1, a2, a3, b1, b2, b3, c1, c2, c3 x is the 3 by 1 matrix containing elements x, y, z and b is the 3 by 1 matrix containing elements b1, d2, d3 ax is equal to b also implies that x is given by a inverse into b Now these system of equations can be written in the matrix form or can be written as ax is equal to b where a is the 3 by 3 matrix containing elements 3, 10, 9, 5, 6, 7, 12, 4, 6 x is the 3 by 1 matrix containing elements s, m, f and b is the 3 by 1 matrix containing elements 420, 360, 420 and we know that x is given by a inverse into b Now we shall find a inverse which is equal to 1 by determinant of a into a joint of a determinant of a is given by which is equal to 3 into 6 into 6 that is 36 minus 7 into 4 that is 28 minus 5 into 10 into 6 that is 60 minus of 4 into 9 that is 36 plus 12 into 10 into 7 that is 70 minus of 9 into 6 that is 54 which is equal to 3 into 36 minus 28 is 8 minus of 5 into 60 minus 36 is 24 plus 12 into 70 minus 54 is 16 3 into 8 is 24 minus of 5 into 24 is 120 plus 12 into 16 is 192 which I am further solving gives 96 so determinant of a is equal to 96 Now we shall find a joint of a which is given by the 3 by 3 matrix containing elements a11, a12, a13, a21, a22, a23, a31, a32, a33, a11 a11 is equal to cofactor of a11 that is 3 and is given by minus 1 raise to power 1 plus 1 into determinant of 6476 which is equal to 1 into 16 to 6 that is 36 minus of 7 into 4 that is 28 a12 is equal to cofactor of a12 that is 5 and is equal to minus 1 raise to power 1 plus 2 into determinant containing elements 10, 9, 4, 6 which is given by minus 1 into 10 into 6 that is 60 minus of 9 into 4 that is 36 which is equal to minus 1 into 24 that is minus of 24 a13 is equal to cofactor of a13 that is 12 and is given by minus 1 raise to power 1 plus 3 into determinant containing elements 10, 9, 6, 7 and is given by 1 into 10 into 7 that is 70 minus of 9 into 6 that is 54 which is equal to 16 so we have 1 into 16 that is 16 a21 is equal to cofactor of a21 that is 10 and is given by minus 1 raise to power 2 plus 1 into determinant containing elements 5, 7, 12, 6 which is equal to minus 1 into 6 into 5 that is 30 minus of 7 into 12 that is 84 which is equal to minus 1 into minus of 54 that is equal to 54 a22 is equal to cofactor of a22 that is 6 and is given by minus 1 raise to power 2 plus 2 into determinant containing elements 3, 9, 12, 6 and is equal to minus 1 raise to the power 4 that is equal to 1 into 3 into 6 that is 18 minus of 9 into 12 that is 108 which is equal to minus 90 and is given by 1 into minus 90 that is minus 90 a23 is equal to cofactor of a23 that is 4 and is given by minus 1 raise to power 2 plus 3 into determinant containing elements 3, 9, 5, 7 which is equal to minus 1 raise to the power 5 that is minus 1 into 3 into 7 that is 21 minus of 9 into 5 that is 45 which is equal to minus 1 into 21 minus 45 is minus 24 that is minus 1 into minus 24 is 24 now a31 is equal to cofactor of a31 that is 9 and is given by minus 1 raise to the power 3 plus 1 into determinant containing elements 5, 6, 12, 4 and is given by minus 1 raise to the power 4 that is 1 into 5 into 4 that is 20 minus of 6 into 12 that is 72 which is equal to 1 into 20 minus 72 is minus 52 that is 1 into minus of 52 is equal to minus of 52 a32 is equal to cofactor of a32 that is 7 and is given by minus 1 raise to power 3 plus 2 into determinant containing elements 3, 10, 12, 4 which is equal to minus 1 raise to the power 5 is minus 1 into 3 into 4 that is 12 minus of 10 into 12 that is 120 which is equal to minus 1 into 12 minus 120 is minus of 108 and is given by minus 1 into minus of 108 that is 108 now a33 is equal to cofactor of a33 that is 6 and is given by minus 1 raise to the power 3 plus 3 into determinant containing elements 3, 10, 5, 6 minus 1 raise to the power 6 is 1 into 3 into 6 that is 18 minus of 10 into 5 that is 50 which is equal to 1 into 18 minus 50 is minus of 32 that is 1 into minus of 32 is equal to minus of 32 therefore a joint of a which is given by the 3 by 3 matrix containing elements a11, a12, a13, a21, a22, a23, a31, a32, a33 is given by a11 is equal to 8, a12 is minus of 24, a13 is given by 16, a21 is equal to 54, a22 is equal to minus of 90, a23 is equal to 24, a31 is given by minus of 52 a32 is given by 108, a33 is given by minus of 32 therefore a joint of a is given by the 3 by 3 matrix containing elements 8 minus 24, 16, 54 minus 90, 24 minus 52, 108 minus 32 a inverse is given by 1 by determinant of a into a joint of a and determinant of a is given by 96 so we have a inverse is equal to 1 by determinant of a that is 96 into a joint of a which is given by the 3 by 3 matrix containing elements 8 minus 24, 16, 54 minus 90, 24 minus 52, 108 minus 32 and we know that x is given by a inverse into b x is equal to a inverse that is 1 by 96 into the 3 by 3 matrix containing elements 8 minus 24, 16, 54 minus 90, 24 minus 52, 108 minus 32 into b that is b is the 3 by 1 matrix containing elements 420, 360, 420 now if we multiply these two matrices then the resultant multiplicant will be of order 3 into 1 so we have 1 by 96 into 3 by 1 matrix containing elements 8 into 420 plus 54 into 360 minus 52 into 420 minus 24 into 420 minus 90 into 360 plus 108 into 420, 16 into 420 plus 24 into 360 minus 32 into 420 which is equal to 1 by 96 into 3 by 1 matrix containing elements 8 into 420 is 3360 plus 54 into 360 is 19,440 minus 52 into 420 is 21,840 minus 24 into 420 is minus of 10,080 minus of 90 into 360 is 32,400 plus 108 into 420 is 35,360 now 16 into 420 is 6720 plus 24 into 360 is 8630 minus of 32 into 420 is 13,430 which is equal to 1 by 96 into 3 by 1 matrix containing elements 960, 2880, 1920 which is equal to the 3 by 1 matrix containing elements 960 divided by 96 that is 10, 2880 divided by 96 that is 30, 1920 divided by 96 that is 20 so we have x is equal to the 3 by 1 matrix containing elements 10, 30, 20 which can be written as the 3 by 1 matrix containing elements SNS is equal to the 3 by 1 matrix containing elements 10, 30, 20 so we can write x is equal to 10, m is equal to 30 and x is equal to 20 hence best way for senior citizens is 10 dollars, for males is 30 dollars and for seniors is 20 dollars which is the required answer this completes our session hope you enjoyed this session