 Hello and welcome to the session. In this session we are going to discuss the following question which says that if A is equal to the 3 by 3 matrix containing elements 1, 2, 2, 1, 3, 3, 1, 4, minus 4, B is equal to the 3 by 3 matrix containing elements 3 minus 2, 0, minus 7 by 8, 6 by 8, 1 by 8, minus 1 by 8, 1 by 4, minus 1 by 8. Find the product of A and B and use it to solve the system of equations given below. x plus y plus z is equal to 30,000. 2x plus 3y plus 4z is equal to 100,000. 2x plus 3y minus 4z is equal to 5,000. Now we are given A is equal to the 3 by 3 matrix containing elements 1, 2, 2, 1, 3, 3, 1, 4, minus 4, and B is equal to the 3 by 3 matrix containing elements 3 minus 2, 0, minus 7 by 8, 6 by 8, 1 by 8, minus 1 by 8, 1 by 4, minus 1 by 8. Now A into B is given by the 3 by 3 matrix containing elements 1, 2, 2, 1, 3, 3, 1, 4, minus 4 into B that is the 3 by 3 matrix containing elements 3 minus 2, 0, minus 7 by 8, 6 by 8, 1 by 8, minus 1 by 8, 1 by 4, minus 1 by 8. Since A and B both have 3 by 3 matrices therefore the resultant multiplicant will be a 3 by 3 matrix containing elements 1 into 3 that is 3 plus 1 into minus 2 that is minus 2 plus 1 into 0 that is 0. Next is 1 into minus 7 by 8 that is minus 7 by 8 plus 1 into 6 by 8 that is plus 6 by 8 plus 1 into 1 by 8 that is plus 1 by 8. Then 1 into minus 1 by 8 that is minus 1 by 8 plus 1 into 1 by 4 that is 1 by 4 plus 1 into minus 1 by 8 that is minus 1 by 8. Next we have 2 into 3 that is 6 plus 3 into minus 2 that is minus 6 plus 4 into 0 that is 0 then 2 into minus 7 by 8 that is minus 14 by 8 plus 3 into 6 by 8 that is 18 by 8 plus 4 into 1 by 8 that is 4 by 8. Next we have 2 into minus 1 by 8 that is minus 2 by 8 3 into 1 by 4 that is plus 3 by 4 plus 4 into minus 1 by 8 that is minus 4 by 8. Next we have 2 into 3 that is 6 plus 3 into minus 2 that is minus 6 minus 4 into 0 is 0. Next 2 into minus 7 by 8 is minus 14 by 8 plus 3 into 6 by 8 is plus 18 by 8 minus 4 into 1 by 8 is minus 4 by 8. Next is 2 into minus 1 by 8 that is minus 2 by 8 3 into 1 by 4 is plus 3 by 4 minus 4 into minus 1 by 8 is plus 4 by 8 which is equal to the 3 by 3 matrix containing elements 3 minus 2 is 1 minus 7 by 8 plus 6 by 8 plus 1 by 8 is equal to 0 minus 1 by 8 plus 1 by 4 minus 1 by 8 is equal to 0. 6 minus 6 plus 0 is 0 minus 14 by 8 plus 18 by 8 plus 4 by 8 is 1 minus 2 by 8 plus 3 by 4 minus 4 by 8 is 0 6 minus 6 plus 0 is 0 minus 14 by 8 plus 18 by 8 minus 4 by 8 is equal to 0 minus 2 by 8 plus 3 by 4 plus 4 by 8 is 1. So, we have a into b is equal to b 3 by 3 matrix containing elements 1 0 0 0 1 0 0 0 1 which is the identity matrix denoted by i. So, we have a b is equal to i that is the identity matrix which implies that a inverse is equal to b and b is the 3 by 3 matrix containing elements 3 minus 2 0 minus 7 by 8 6 by 8 1 by 8 minus 1 by 8 1 by 4 minus 1 by 8. We are given the following system of equations x plus y plus z is equal to 30,000 2x plus 3y plus 4z is equal to 100,000 2x plus 3y minus 4z is equal to 5000. Now, we can write these system of equations in the matrix form as the 3 by 3 matrix containing elements 1 2 2 1 3 3 1 4 minus 4 into the 3 by 1 matrix containing elements x y z is equal to the 3 by 1 matrix containing elements 30,000 100,000 5000 and we can also write it as a x is equal to c where a is the 3 by 3 matrix containing elements 1 2 2 1 3 3 1 4 minus 4 x is the 3 by 1 matrix containing elements x y z and c is the 3 by 1 matrix containing elements 30,000 100,000 5000. Now, a x is equal to c implies that x is equal to a inverse of c and we know that a inverse is given by the 3 by 3 matrix containing elements 3 minus 2 0 minus 7 by 8 6 by 8 1 by 8 minus 1 by 8 1 by 4 minus 1 by 8. So, we write x is equal to a inverse that is the 3 by 3 matrix containing elements 3 minus 2 0 minus 7 by 8 6 by 8 1 by 8 minus 1 by 8 1 by 4 minus 1 by 8 into c that is the 3 by 1 matrix containing elements 30,000 100,000 5000 into 3 by 1 matrix containing elements 30,000 100,000 5000 since a inverse is the 3 by 3 matrix and c is the 3 by 1 matrix their resultant multiplicant will be a 3 by 1 matrix containing elements 3 into 30,000 plus minus 7 by 8 into 100,000 plus minus 1 by 8 into 5000 next we have minus 2 into 30,000 plus 6 by 8 into 100,000 plus 1 by 4 into 5000 and then 0 into 30,000 plus 1 by 8 into 100,000 minus 1 by 8 into 5000 which is equal to the 3 by 1 matrix containing elements 720,000 minus 700,000 minus 5000 by 8 minus 480,000 plus 600,000 plus 10,000 by 8 and then 100,000 minus 5000 by 8 which on server solving this a 3 by 1 matrix containing elements 1,875 16,250 11,875 so x is given by a 3 by 1 matrix containing elements 1,875 16,250 11,875 or we can say x is given by the 3 by 1 matrix containing elements x, y, z is equal to the 3 by 1 matrix containing elements 1,875 16,250 11,875 therefore x is given by 1875 y is given by 16,250 and z is equal to 11,875 which is the required answer this completes our session hope you enjoyed this session