 Hello and welcome to the session, I am Deepika here. Let's discuss the question. Examine the consistency of the system of equations 3x-y-2z is equal to 2, 2y-z is equal to minus 1, 3x-5y is equal to 3. Let's start the solution. The given system of equations is minus 5 minus 2z is equal to 2, 2y-z is equal to minus 1, 3x-5y is equal to 3. The system of equations can be written in the form x is equal to b, where x is equal to minus 1, minus 2, 0, 2, minus 1, 3, minus 5, 0. x is equal to x-y-z and b is equal to minus 1, 3. We see that determinant a is equal to 3 into 0 minus 5 plus 1 into 0 plus 3 and minus 2 into 0 minus 6. Which is equal to minus 15 plus 3 plus 12 which is equal to 0. This implies a is a singular matrix. As we will calculate adjoint a into b. Now for adjoint a we will find co-factors of all the elements. Now co-factor of 3 is equal to minus 1 raise to power 1 plus 1 into 0 minus 5 which is equal to minus 5. Co-factor of minus 1 is equal to minus 1 raise to power 1 plus 2 into 0 plus 3. That is co-factor of minus 1 raise to power 1 plus 2 into 0 plus 3 which is equal to 3. Again co-factor of minus 2 is equal to minus 1 raise to power 1 plus 3 into 0 minus 6. We can find the co-factors of all the elements as. Now matrix form y is equal to minus 5 minus 3 minus 6, 10, 6, 12, 5, 3, 6. Therefore adjoint a is equal to transpose of this. It is minus 5 minus 3 minus 6, 10, 6, 12, 5, 3, 6 which is equal to minus 5 minus 3 minus 6, 10, 6, 12, 5, 3, 6. Therefore adjoint a into v is equal to minus 5 minus 3 minus 6, 10, 6, 12, 5, 3, 6 into 2 minus 1, 3. This is equal to minus 5 into 2 minus 10, 10 into minus 1 minus 10 plus 5 into 3, 15. Again minus 3 into 2 minus 6, 6 into minus 1 minus 6 plus 3 into 3, 9. Again minus 6 into 2 minus 12 minus 12 plus 18 which is equal to 5 minus 3 and minus 6. This implies adjoint a into b is not equal to 0. Adjoint a into a 0 matrix implies the system of equations is inconsistent. Concern for the above system of equations is inconsistent. I hope the question is clear to you. Bye and have a good day.