 Hello and welcome to this session. I am Ashapa and I am there to help you with the following problem. Using properties of the terminus prove that this is the terminus alpha, alpha squared, beta plus gamma, beta, beta squared, alpha plus gamma, gamma, gamma squared, alpha plus beta is equal to beta minus gamma, gamma minus alpha, alpha minus beta multiplied by alpha plus beta plus gamma. Now let us write the solution. Let us consider LHS which is equal to alpha, alpha squared, beta plus gamma, beta, beta squared, gamma plus alpha, gamma, gamma squared, alpha plus beta. Now applying R1 tends to R1 minus R2 and R2 tends to R2 minus R3. We get alpha minus beta, alpha squared minus beta squared, beta plus gamma minus gamma minus alpha, beta minus gamma, beta squared minus gamma squared, gamma plus alpha minus alpha minus beta, gamma, gamma squared, alpha plus beta. Now we see here this gets cancelled, this gets cancelled. Now which is equal to alpha minus beta, alpha minus beta into alpha plus beta into beta minus alpha, beta minus gamma, beta minus gamma into beta plus gamma into gamma minus beta, gamma, gamma squared, alpha plus beta which is equal to now taking alpha minus beta and beta minus gamma common. So we are left with 1 alpha plus beta minus 1 beta plus gamma minus 1 gamma, gamma squared, alpha plus beta. So now applying R1 tends to sorry C1 tends to C1 plus C3. So we get it is equal to alpha minus beta, beta minus gamma multiplied by 1 minus 1 is 0, alpha plus beta is equal to minus 1 as it is. Now again 1 minus 1 is 0, beta plus gamma as it is and minus 1 as it is. Now gamma plus alpha plus beta which is equal to alpha plus beta plus gamma, gamma squared as it is and alpha plus beta as it is. Now from this column taking alpha plus beta plus gamma common we are left with 0 alpha plus beta minus 1 0 beta plus gamma minus 1 1 gamma squared alpha plus beta. Now which is equal to alpha minus beta beta minus gamma alpha plus beta plus gamma multiplied by now expanding through 1. So eliminating this row and this column we get minus of alpha plus beta plus beta plus gamma which is equal to alpha minus beta beta minus gamma alpha plus beta plus gamma into minus alpha minus beta plus beta plus gamma. Now here we see that this beta and beta gets cancelled. So we get alpha minus beta beta minus gamma and this is as gamma minus alpha into alpha plus beta plus gamma. This is our required answer which is equal to RHS. Therefore LHS is equal to RHS hence proved. I hope you understood the problem. Bye and have a nice day.