 Hi and welcome to our session. It is just as the following question. The question says using properties of determinants proof that this determinant is equal to alpha minus beta into beta minus gamma into gamma minus alpha. This is the solution. We will first consider left inside. This is equal to determinant of 1 1 1 alpha beta gamma beta gamma alpha alpha beta. Now we will apply column operation C2 goes to C2 minus 3 and C3 goes to C3 minus 1. So by applying this column operation we get determinant of 1 alpha beta gamma 0 beta minus gamma alpha into gamma minus beta 0 gamma minus alpha beta into alpha minus gamma. Now this is equal to 1 into determinant of beta minus gamma alpha into gamma minus beta gamma minus alpha beta into alpha minus gamma minus 0 into determinant of alpha beta gamma minus alpha beta into alpha minus gamma plus 0 into determinant of alpha beta gamma beta minus gamma alpha into gamma minus beta is equal to 1 into determinant of beta minus gamma gamma minus alpha alpha into gamma minus beta beta into alpha minus gamma. Now as both these terms finishes. So we are left with only this expression. Now this is equal to beta minus gamma into gamma minus alpha. We have taken both these common from this determinant. So in determinant we are left with 1 minus alpha minus beta beta into minus beta. Hence we have to write inside.