 Hello and welcome to the session, I am Deepika here. Let's discuss the question. The question says, Solve system of linear equations using matrix method. 2x plus y plus z is equal to 1. x minus 2y minus z is equal to 3 by 2. 3y minus 5z is equal to 9. Let's start the solution. The given system of equations can be written in the form ax is equal to b where a is equal to 2 1 1 1 minus 2 minus 1 0 3 minus 5 is equal to x y z and b is equal to 1 3 by 2 9. Now determinant a is equal to 2 into 10 plus 3 minus 1 into minus 5 minus 0 plus 1 into 3 minus 0 or 3 plus 0 1 and the same thing. This is equal to 2 into 13 plus 5 plus 3 this is equal to 34 which is not equal to 0. This implies a is non-singular and so its inverse exists. Now we will find out a inverse. For a inverse we will first find out adjoint a and for adjoint a we have to find out the co-factors of each and every element. Now co-factor of 2 is equal to minus 1 raise to power 1 plus 1 into 10 plus 3 that is minus 1 raise to power 1 plus 1 into 10 plus 3 which is equal to 13. Again co-factor of 1 is equal to minus 1 raise to power 1 plus 2 into 5 plus 0 that is co-factor of minus 1 raise to power 1 plus 2 into minus 5 plus 0 which is equal to 5. Again co-factor of 1 is equal to minus 1 raise to power 1 plus 3 because it is the first row and the third column into 3 plus 0. Again co-factor of 1 is equal to minus 1 raise to power 1 plus 3 into 3 plus 0 which is equal to 3. Similarly we can find out the co-factors of second row and third row. So hence we have the co-factors of the other elements of matrix. Now matrix formed by the co-factors is equal to 13 5 3 8 minus 10 minus 6 1 3 minus 5. Therefore now adjoint A is equal to transpose of this matrix which is equal to 13 5 3 8 minus 10 minus 6 1 3 minus 5. Now this is equal to 1 over determinant A into adjoint A. This implies A inverse is equal to adjoint A is our 13 5 3 8 minus 10 minus 6 1 3 minus 5 into 1 over determinant A which is equal to 1 over 34. Now A x is equal to B implies inverse A x is equal to A inverse B by pre multiplying both sides by. So we get A inverse into A x is equal to A inverse B that is I x is equal to A inverse B which implies this is equal to now A inverse is 1 over 34 into 8 minus 10 minus 6 1 3 minus 5 into B which is 1 3 by 2 and 9. Now we will solve this we get this is equal to 1 over 34 into 13 into 1 13 plus 8 into 3 by 2 plus 1 into 9. That is 9. Again 5 into 1 plus minus 10 into 3 by 2 plus 3 into 9 27. Again we have 3 into 1 plus minus 6 into 3 by 2 plus minus 45. So this is equal to we get this is equal to 1 over 34 into 13 plus 12 plus 9 which is equal to 34 and 5 minus 15 plus 27 this is equal to 17 and 3 minus 9 minus 45 this is equal to minus 51. This is again equal to 1 1 by 2 and minus 3 by 2 this implies our xyz is equal to 1 1 by 2 minus 3 by 2 on equating the corresponding elements we get x is equal to 1 y is equal to 1 by 2 and z is equal to minus 3 by 2 hence we have solved the above system of linear equations using matrix method and our answer is x is equal to 1 y is equal to 1 by 2 and z is equal to minus 3 by 2 I hope you have enjoyed this session bye and take care