 Hello and welcome to the session. In this session we discussed the following question which says, by using the properties of determinants, show that determinant with elements a plus b plus c minus c minus b minus c, a plus b plus c minus a minus b minus a, a plus b plus c is equal to 2 into a plus b into b plus c into c plus a. Let's move on to the solution now. We need to show that determinant with elements a plus b plus c minus c minus b minus c, a plus b plus c minus a minus b minus a, a plus b plus c is equal to 2 into a plus b into b plus c into c plus a. So we will consider the LHS and we take let delta be equal to the LHS that is determinant with elements a plus b plus c minus c minus b minus c, a plus b plus c minus a minus b minus a, a plus b plus c. Now applying C1 goes to C1 plus C2 we get delta is equal to, determinant with elements a plus b minus c minus b, a plus b, a plus b plus c minus a minus a plus b minus a, a plus b plus c. That is the elements of column C2 and C3 remain same and elements of C1 are given by C1 plus C2. Now taking out common factor a plus b from C1 we get delta is equal to a plus b into determinant with elements 1 minus c minus b, 1 a plus b plus c minus a minus 1 minus a, a plus b plus c. Now next applying C2 goes to C2 plus C3 we get delta is equal to a plus b into determinant with elements 1 minus b plus c minus b then 1 b plus c minus a minus 1 b plus c, a plus b plus c. Now elements of C1, C3 remain same and elements of C2 are given by C2 plus C3. Now next taking out common factor b plus c from C2 we get delta is equal to a plus b into b plus c into determinant with elements 1 minus 1 minus b, 1, 1 minus a minus 1, 1, a plus b plus c. Now expanding the determinant along R1 we get delta is equal to a plus b into b plus c into, now consider the first element of R1 that is 1 into, 1 into a plus b plus c minus 1 into minus a then minus consider the second element of R1 that is minus 1 this into 1 into a plus b plus c minus minus 1 into minus a. Now plus the third element of R1 that is minus b into 1 into 1 minus minus 1 into 1 thus we get delta is equal to a plus b into b plus c into a plus b plus c, plus a plus 1 into a plus b plus c minus a minus b into 1 plus 1. So this further gives us delta is equal to a plus b into b plus c into 2 a plus b plus c here a and minus a cancels so we have plus b plus c minus 2 b from here we get delta is equal to a plus b into b plus c into 2 a plus b plus c plus b plus c minus 2 b that is we have delta is equal to a plus b into b plus c into 2 into a plus c plus 2 b minus 2 b. Now this 2 b and minus 2 b cancels so we have delta is equal to 2 into a plus b into b plus c into a plus c or c plus a and this is equal to the RHS thus we have the LHS is equal to the RHS. So hence proved this complete C session hope you have understood the solution for this question.