 Hello friends, welcome to the session, I am Malka. Let's discuss the given question. Verify that A into a joint of A equal to, a joint of A into A equal to determinant of A into I. A exercises 3 and 4. As the given matrix is 2, 3, minus 4, minus 6. Now let's begin with the solution. We have to verify that A into a joint of A equal to a joint of A into A equal to determinant of A into I. And to prove this, we have to first find out a joint of A and a joint of A is the transport of the matrix formed by the co-factors. So first of all we have to find the co-factors of the elements of the given matrix. So our given matrix is, let A equal to matrix 2, 3, minus 4, minus 6. Now we will find the co-factors of the elements of the given matrix. So co-factor of the elements A ij is denoted by A A ij equal to minus 1 to the power i plus j into M ij, where M ij is the minor of the element that is A ij. Now we will write the co-factor of the elements of the matrix A. So co-factor A 11 that is 2 is equal to, is denoted by A 11 equal to minus 1 to the power 1 plus 1 into M 11. And we know that minor of first row and first column that is 2 is obtained by deleting the first row and first column that is M 11 is minus 6. So it will be written as minus 1 to the power 2 into minus 6. It is equal to, now minus 1 to the power 2 is plus 1 into minus 6 is minus 6. Therefore co-factor of 2 is minus 6. Now co-factor of A 12, 3 this is denoted by A 12 equal to minus 1 to the power 1 plus 2 into M 12. Now we have to find M 12. You can see from the matrix that M 12 that is first row and second column. So we have to find the co-factor of 3. So we will delete the first row and second column. So M 12 is minus 4. So we will write minus 1 to the power 3 into minus 4. So this is equal to plus 4. Now we will write the co-factor of 2 1 that is minus 4 is denoted by A 21 equal to minus 1 to the power 2 plus 1 into M 21. Now we have to find minor that is M 21. So we have to find the minor of minus 4. So we will delete the first column and the second row. So it is 3. So therefore we will write this is equal to minus 1 to the power 3 into 3. This is equal to minus 3. Similarly we will write the co-factor of A 22 that is minus 6 equal to A 22 equal to this is denoted by capital A 22 which is equal to minus 1 to the power 2 plus 2 into M 22. Now we find M 22. So this is equal to minus 1 to the power 4. Now minor of minus 6 will be we will delete the second row and second column. So it will be 2. So we get minus 1 into 2. So this is equal to 2. Now therefore matrix formed by the co-factors is A 6 4 minus 3 and the adjoint of A and we know that adjoint of A is the transpose of the matrix formed by the co-factors. So that is minus 6 4 minus 3 to transpose. This is equal to minus 6 4 minus 3. Since we know that in the transpose of the matrix we interchange the rows and columns. So adjoint of A is minus 6 minus 3 4 and 2. Now we will find A into adjoint of A and we know that the matrix A is 2 3 minus 4 minus 6 and adjoint of A is minus 6 minus 3 4 and 2. This is equal to now we multiply both the matrices we get 2 into minus 6 plus 3 into 4 then 2 into minus 3 plus 3 into 2. Similarly for the second row minus 4 into minus 6 plus minus 6 into 4 and minus 4 into minus 3 plus minus 6 into 2. So this is equal to minus 12 plus 12 minus 6 plus 6 plus 24 minus 24 plus 12 and minus 12. So we see that the obtain matrix is 0 0 0 and 0. So this is the value of the multiplication of A into adjoint of A. Now we will find adjoint of A into A. So we know that adjoint of A is minus 6 minus 3 4 2 and the value of the matrix A is 2 3 minus 4 minus 6. Now we will multiply both the matrices and we get minus 12 plus 12 minus 18 plus 18 8 minus 8 12 minus 12. So this is equal to 0 0 0 0. So adjoint of A into A is the matrix 0 0 0 0. Now we will find the determinant of A it is equal to minus 12 plus 12 this is equal to 0. Therefore determinant of A into I equal to 0 into identity matrix that is 1 0 0 1. So this is equal to 0 0 0 0. Now we see that we have to verify that A into adjoint of A equal to adjoint of A into A equal to determinant of A into I. We have already calculated the value of A into adjoint of A adjoint of A into A and determinant of A into I and we see that the values of all the three terms is the matrix 0 0 0 0. Therefore we can say that A into adjoint of A equal to adjoint of A into A equal to determinant of A into I. Hence verify into adjoint of A equal to adjoint of A into A equal to determinant of A into I. So hope you understood the solution and enjoyed the session. Goodbye and take care.