 Hello and welcome to the session. Let us discuss the following question which says using properties of determinants prove the following that is the determinant of order 3 by 3 with elements alpha beta gamma alpha square beta square gamma square beta plus gamma gamma plus alpha alpha plus beta is equal to alpha minus beta into beta minus gamma into gamma minus alpha into alpha plus beta plus gamma. Let's see its solution. We are given the determinant of order 3 by 3 with elements alpha beta gamma alpha square beta square gamma square beta plus gamma gamma plus alpha alpha plus beta. So let us assume that this determinant is equal to delta. So now we need to show that delta is equal to alpha minus beta into beta minus gamma into gamma minus alpha into alpha plus beta plus gamma. Now first of all let us apply the operation r1 changes to r1 plus r3. So applying r1 changes to r1 plus r3 we get delta is equal to the determinant of order 3 by 3 with elements alpha plus beta plus gamma beta plus gamma plus alpha gamma plus alpha plus beta alpha square beta square gamma square beta plus gamma gamma plus alpha alpha plus beta. Now in the first row alpha plus beta plus gamma is common in all the elements. So taking common factor alpha plus beta plus gamma from r1 we have delta is equal to alpha plus beta plus gamma into the determinant of order 3 by 3 with elements 1 1 1 alpha square beta square gamma square beta plus gamma gamma plus alpha alpha plus beta. Now all the elements in first row are 1. So applying c2 changes to c2 minus c1 and c3 changes to c3 minus c1. We get delta is equal to alpha plus beta plus gamma into the determinant of order 3 by 3 with elements. Here the elements of c1 will be as it is that is 1 alpha square beta plus gamma. Now c2 changes to c2 minus c1. So the elements of c2 will be 1 minus 1 that is 0 beta square minus alpha square gamma plus alpha minus beta minus gamma. Now c3 changes to c3 minus c1. So the elements of c3 will be 1 minus 1 that is 0 gamma square minus alpha square alpha plus beta minus beta minus gamma. From this we get delta is equal to alpha plus beta plus gamma into the determinant of order 3 by 3 with elements 1 0 0 alpha square beta plus alpha into beta minus alpha gamma plus alpha into gamma minus alpha beta plus gamma alpha minus beta alpha minus gamma. Now from c2 let us take alpha minus beta is common and from c3 let us take gamma minus alpha as common. So taking common factor alpha minus beta and gamma minus alpha from c2 and c3 respectively we have delta is equal to alpha plus beta plus gamma into alpha minus beta into gamma minus alpha into the determinant of order 3 by 3 with elements 1 0 0 alpha square minus of beta plus alpha gamma plus alpha beta plus gamma 1 minus 1. Now in this determinant in the first row two elements are 0. So we will expand this determinant along r1 plus expanding the determinant along r1 we get delta is equal to alpha plus beta plus gamma into alpha minus beta into gamma minus alpha into 1 into minus beta minus alpha into minus 1 minus 1 into gamma plus alpha and this will be equal to alpha plus beta plus gamma into alpha minus beta into gamma minus alpha into beta plus alpha minus gamma minus alpha that is alpha plus beta plus gamma into alpha minus beta into gamma minus alpha into beta minus gamma. Thus delta is equal to alpha minus beta into beta minus gamma into gamma minus alpha into alpha plus beta plus gamma hence proved with this we finished this session hope you must have enjoyed it goodbye take care and have a nice day.