 Hello friends, welcome to the session. I am Alka. We are going to discuss determinants. Our question is verify A into adjoint of A equal to adjoint of A into A equal to determinant of A into I in exercises 3rd and 4th. Our 4th exercise is matrix 1 minus 1 2 3 0 minus 2 1 0 3. Now let's start with the solution. We are given A equal to matrix 1 minus 1 2 3 0 minus 2 1 0 3. Now we will find the co-factor of every element of matrix A. Co-factor of 1 that is A 1 1 equal to minus 1 to the power 1 plus 1 and 0 plus 0 which is equal to 0. Co-factor of minus 1 that is A 1 2 equal to minus 1 to the power 1 plus 2 into 3 3 is a 9 and plus 2 which is equal to minus 11. Similarly, we will write the co-factor of other elements. Co-factor of 2 equal to 0. Co-factor of 3 equal to 3. Co-factor of 0 is 1. Co-factor of minus 2 is minus 1. Co-factor of 1 is 2. Co-factor of 0 is 8. Co-factor of 3 is 3. Therefore, the matrix formed by the co-factors is matrix 0 minus 11 0 3 1 minus 1 2 8 3. Now we will find adjoint of A which is transpose of the matrix formed by the co-factors that is 0 minus 11 0 3 1 minus 1 2 8 3 transpose. This is equal to 0 minus 11 0 3 1 minus 1 2 8 3. This is the value of adjoint of A. Now we will find A into adjoint of A which is equal to matrix 1 minus 1 2 3 0 minus 2 1 0 3 into matrix adjoint of A that is 0 3 2 minus 11 1 8 0 minus 1 3. This is equal to 0 plus 11 plus 0 3 minus 1 minus 2 2 minus 8 plus 6 0 minus 0 minus 0 9 plus 0 plus 2 6 plus 0 minus 6 0 minus 0 plus 0 3 plus 0 minus 3 2 plus 0 plus 9. This is equal to 11 0 0 0 11 0 0 0 11. Now we will find adjoint of A into A which is equal to matrix 0 3 2 minus 11 1 8 0 minus 1 3 into matrix 1 minus 1 2 3 0 minus 2 1 0 3. This is equal to 0 plus 9 plus 2 0 plus 0 plus 0 0 minus 6 plus 6 minus 11 plus 3 plus 8 11 plus 0 plus 0 minus 22 minus 2 plus 24 0 minus 3 plus 3 0 plus 0 plus 0 and 0 plus 2 plus 9. This is equal to 11 0 0 0 11 0 0 0 11. Now we will find the determinant of A that is 1 into 0 plus 0 plus 1 into 9 plus 2 plus 2 into 0 plus 0 which is equal to 11. Now we will find the value of determinant of A into I which is equal to 11 into matrix I which is 1 0 0 0 1 0 0 0 1 which is equal to matrix 11 0 0 0 11 0 0 0 11. Thus we see that adjoint A into adjoint of A equal to adjoint of A into A equal to determinant of A into I. So, it is verified that these three terms are equal to each other. Hope you understood the solution and enjoyed the session. Goodbye and take care.