 Hello and welcome to the session. In this session we are going to discuss the following question which says that use matrix theory to examine the following system of equations for consistency or inconsistency. a. 4x plus 6y is equal to 9. 2x plus 3y is equal to 11. b. x minus 2y is equal to 5. 2x minus 4y is equal to 10. If we are given two simultaneous equations in two variables x and y that is a1x plus b1y is equal to c1 and a2x plus b2y is equal to c2 then these system of equations can be written in the matrix form as ax is equal to b where a is equal to the 2 by 2 matrix containing elements a1 a2 b1 b2 x is the 2 by 1 matrix containing elements xy and b is the 2 by 1 matrix containing elements c1 c2 then if a joint of a into b is not equal to 0 then the system of equations is inconsistent and has no solution and if a joint of a into b is equal to 0 then the system of equations is consistent and has infinite number of solutions with this key idea we shall proceed with the solution. We are given the following system of equations that is 4x plus 6y is equal to 9 and 2x plus 3y is equal to 11. Now representing the system of equations by the matrix equation is equal to b we have 2 by 2 matrix containing elements 4th to 6th 3 into 2 by 1 matrix containing elements xy and b is the 2 by 1 matrix containing elements 911 where a is equal to the 2 by 2 matrix containing elements 4th to 6th 3 x is the 2 by 1 matrix containing elements xy and b is the 2 by 1 matrix containing elements 911 and determinant of a is given by 4 into 3 by 12 minus of 2 into 6 by 12 which is equal to 0 Now determinant of a is equal to 0 therefore a is singular therefore i will use the system has no solution and infinite number of solutions Now we find a joint of a into b a is equal to the 2 by 2 matrix containing elements 4th to 6th 3 and we can find the a joint of a by interchanging the entries on the leading diagonal I am changing the sign of the entries on the other diagonal so a joint of a is equal to the 2 by 2 matrix containing elements 3 minus 2 minus 6 4 therefore a joint of a into b is equal to a joint of a is the 2 by 2 matrix containing elements 3 minus 2 minus 6 4 and b is given by the 2 by 1 matrix containing elements 911 That is 2 by 1 matrix containing elements 911 a joint of a is the 2 by 2 matrix and b is the 2 by 1 matrix so the resultant multiplicant will be a 2 by 1 matrix given by 3 into 9 plus minus of 6 into 11 minus 2 into 9 plus 4 into 11 which is equal to the 2 by 1 matrix containing elements 27 minus 66 minus 18 plus 34 which is equal to the 2 by 1 matrix containing elements minus 39 26 which is not equal to 0 therefore a joint of a into b is not equal to 0 and from the key idea we know that if the joint of a into b is not equal to 0 then the system of equations is inconsistent and has no solution therefore the given system of equations is inconsistent and has no solution we have another system of equations that is x minus 2y is equal to 5 2x minus 4y is equal to 10 we mark the equation x minus 2y is equal to 5 as 1 and 2x minus 4y is equal to 10 as 2 now representing the system the matrix equation ax is equal to b we have a 2 by 2 matrix containing elements 1 2 minus 2 minus 4 into 2 by 1 matrix containing elements xy is equal to 2 by 1 matrix containing elements 5 10 where a is equal to the 2 by 2 matrix containing elements 1 2 minus 2 minus 4 x is the 2 by 1 matrix containing elements xy and b is the 2 by 1 matrix containing elements 5 10 now determinant of a is given by 1 into minus 4 that is minus 4 minus of 2 into minus 2 that is minus 4 which is equal to minus 4 plus 4 that is 0 so determinant of a is equal to 0 therefore a is singular therefore either the given system has no solution or an infinite number of solutions now we find a joint of a into b and a is given by the 2 by 2 matrix containing elements 1 2 minus 2 minus 4 and we can obtain the adjoint of a by interchanging the entries on the leading diagonal and changing the sign of the entries on the other diagonal so adjoint of a is given by the 2 by 2 matrix containing elements minus 4 minus 2 2 1 therefore adjoint of a into b is given by the 2 by 2 matrix minus 4 minus 2 2 1 into b and b is given by the 2 by 1 matrix containing elements 5 10 that is 2 by 1 matrix containing elements 5 10 now adjoint of a is a 2 by 2 matrix and b is a 2 by 1 matrix so the resultant multiplicant will be 2 by 1 matrix containing elements minus 4 into 5 plus 2 into 10 minus 2 into 5 plus 1 into 10 which is equal to the 2 by 1 matrix containing elements minus 20 plus 20 minus 10 plus 10 which is equal to the 2 by 1 matrix containing elements 0 0 that is 0 so adjoint of a into b is equal to 0 and from the key idea we know that if adjoint of a into b is equal to 0 then the system of equations is consistent and has infinite number of solutions therefore the given system of equations is consistent and has infinite number of solutions now put y is equal to k in the first equation the first equation is x minus 2 y is equal to 5 so we have x minus 2 k is equal to 5 which implies that x is equal to 5 plus 2 k now putting the values of x and y in equation 2 we get equation 2 is 2 x minus 4 y is equal to 10 so we have 2 into x that is 5 plus 2 k minus 4 y that is minus 4 k is equal to 10 which implies that 2 into 5 that is 10 plus 2 into 2 k that is 4 k minus 4 k is equal to 10 10 is equal to 10 which is true hence the given system of equations infinite number of solutions given by x is equal to 5 plus 2 k and y is equal to k which is the required answer this completes our session hope you enjoyed this session