 Hello and welcome to the session. Let us understand the following problem today. Let A is equal to the determinant 1 minus 2, 1 minus 2, 3, 1, 1, 1 and 5. Verify that first adjoint A the whole inverse is equal to adjoint of A inverse and second A inverse of whole inverse is equal to A. Now let us write the solution. Given to us is A which is equal to 1 minus 2, 1 minus 2, 3, 1, 1, 1 and 5. Now let us first find the determinant of A which is equal to 1 multiplied by 15 minus 1 plus 2 multiplied by minus 10 minus 1 plus 1 multiplied by minus 2 minus 3. Now solving it further we get 14 minus 22 minus 5 which is equal to minus 13. So we see that it is not equal to 0 which implies A inverse exists. Now let us find the co-factors. So A11 is equal to minus 1 to the power 1 plus 1 15 minus 1 which is equal to 14. Then A12 is equal to minus 1 to the power 1 plus 2 minus 10 minus 1 which is equal to 11. A13 is equal to minus 1 to the power 1 plus 3 minus 2 minus 3 is equal to minus 5. A21 is equal to minus 1 to the power 2 plus 1 multiplied by minus 10 minus 1 which is equal to 11. Now A22 is equal to minus 1 to the power 2 plus 2 multiplied by 5 minus 1 which is equal to 4. A23 is equal to minus 1 to the power 2 plus 3 multiplied by 1 plus 2 which is equal to minus 3. Now A31 is equal to minus 1 to the power 3 plus 1 which is multiplied by minus 2 minus 3 which is equal to minus 5. A32 is equal to minus 1 to the power 3 plus 2 1 plus 2 is equal to minus 3. And finally A33 is equal to minus 1 to the power 3 plus 3 multiplied by 3 minus 4 is equal to minus 1. Now let us write the matrix formed by the cofactors is equal to 1411 minus 5114 minus 3 minus 5 minus 3 minus 1. Therefore we get a joint of A is equal to transpose of this matrix so it is equal to 1411 minus 5114 minus 3 minus 5 minus 3 minus 1. This adjoint A we have obtained by entertaining rows and columns of this matrix. Now since we have to find adjoint A the whole inverse so we will need to find again the adjoint of adjoint A. So let us proceed first let us find determinant of adjoint A which is equal to 14 multiplied by minus 4 minus 9 minus 11 minus 11 minus 15 minus 5 minus 33 plus 20. Which is equal to 14 multiplied by minus 13 minus 11 multiplied by minus 26 minus 5 multiplied by minus 13 which is equal to taking 13 common we get minus 14 plus 22 plus 5. So we get here 13 multiplied by 13 which is equal to 169 which is not equal to 0 therefore A adjoint of A inverse exists. Now let us find the cofactors. Now similarly like the previous one like these cofactors we will find the cofactors for this matrix. So now let us see the cofactors C11 is equal to minus 1 to the power 1 plus 1 minus 4 minus 9 which is equal to minus 13. C12 is equal to minus 1 to the power 1 plus 2 minus 11 minus 5 is equal to minus 26 C13 is equal to minus 1 to the power 1 plus 3 multiplied by minus 33 plus 20 is equal to minus 13. C21 is equal to minus 1 to the power 2 plus 1 minus 11 minus 15 is equal to 26. C22 is equal to minus 1 to the power 2 plus 2 multiplied by minus 14 minus 25 is equal to minus 13. C23 is equal to minus 1 to the power 2 plus 3 multiplied by minus 42 plus 55 is equal to minus 13. Similarly C31 is equal to minus 13, C32 is equal to minus 13, C33 is equal to minus 65. Now let us write the matrix formed by the cofactors which is equal to minus 13, 26, minus 13, 26, minus 39, minus 13, minus 13, minus 16. Now finding the adjoint of adjoint A which is equal to minus 13, 26, minus 13, 26, minus 39, minus 13, minus 13, minus 13, minus 65. Therefore adjoint A the whole inverse is equal to 1 by adjoint A determinant multiplied by adjoint of adjoint A which is equal to 1 by 169 multiplied by determinant minus 13, 26, minus 13, 26, minus 39, minus 13, minus 13, minus 13, minus 65. Which is nothing but equal to minus 13 by 169, 26 by 169, minus 13 by 169, 26 by 169, minus 39 by 169, minus 13 by 169, minus 13 by 169, minus 13 by 169, minus 13 by 169, minus 169, minus 165 by 169. Now let us name this as 1. Now A inverse is equal to 1 by determinant of A multiplied by adjoint A which is equal to 1 by minus 13 multiplied by determinant 14, 11, minus 5, 11, 4, minus 3, minus 5, minus 3, minus 1. Which is equal to determinant minus 14 by 13, minus 11 by 13, 5 by 13, minus 11 by 13, minus 4 by 13, 3 by 13, 5 by 13, 3 by 13, 1 by 13. Now similarly we can find the cofactors for this matrix and find adjoint of A inverse, which comes out to be adjoint of A inverse is equal to minus 13 by 169, 26 by 169, minus 13 by 169, minus 13 by 169, minus 13 by 169, minus 13 by 169, minus 165 by 169. minus 13 by 169, minus 13 by 169, minus 13 by 169, minus 65 by 169. This matrix we have obtained by finding the co-factors, so this is equal to a joint of A, the whole inverse that is from 1 we got, hence we have proved the first, now proceeding further we get, now we have A inverse as this, minus 14 by 13, minus 11 by 13, 5 by 13, minus 11 by 13, minus 4 by 13, 3 by 13, 5 by 13, 3 by 13, 1 by 13. Now let us find the determinant of A inverse which is equal to minus 14 by 13 multiplied by minus 4 by 169, minus 9 by 169, plus 11 by 13 multiplied by minus 11 by 169, minus 15 by 169, plus 5 by 13 multiplied by minus 33 by 169, plus 20 by 169. Now which is equal to taking 13 by 13 multiplied by 169 common, so we get here 14 minus 22 minus 5, this one gets cancels, so we are left with 1 by 169 multiplied by minus 13, now here this gets cancels with 13, so we get minus 1 by 1, 13, so therefore A inverse of inverse is equal to 1 by determinant of A inverse multiplied by a joint of A inverse which is equal to 1 by minus 1 by 13 multiplied by this matrix with minus 13 by 169, 26 by 169, minus 13 by 169, 26 by 169, minus 36 by 169, minus 39 by 169, minus 13 by 169, minus 13 by 169, minus 65 by 169, which on solving is equal to minus 13 multiplied by 13 by 169 multiplied by this matrix with minus 1, 2, minus 1, 2, minus 3, minus 1, minus 1, minus 5, now here this gets cancels with 13 and this gets cancels with again 13, so we are left with this matrix 1 minus 2, 1 minus 2, 3, 1, 1, 1, 5 which is nothing but equal to A, therefore A inverse of whole inverse is equal to A, hence we have proved second part also, hence proved both first and second. I hope you understood this problem, bye and have a nice day.