 Hello and welcome to the session. I am the picker here. Let's discuss a question by using properties of determinants show that Determinant a minus b minus c 2 a 2 a 2 b b minus c minus a 2 b 2 c 2 c c minus a minus b is equal to a plus b plus c whole q Let's start the solution we have on left-hand side equal to a minus b minus c 2 b 2 c 2 a b minus c minus a 2 c 2 a 2 b c minus a minus b by applying R1 goes to R1 plus R2 plus R3 We get our left-hand side is equal to now R1 goes to R1 plus R2 plus R3 that is a minus b minus c plus 2 b plus 2 c is Equal to a plus b plus c again 2 a plus b minus c minus a plus 2 c is Same as a plus b plus c again here. We get 2 a plus 2 b plus c minus a Minus b is equal to a plus b plus c R2 and R3 are same Again method is same take a plus b plus c is common in R1. So by taking out by taking out factor a plus b plus c common from R1 we get our left-hand side is equal to a plus b plus c into 1 1 1 2 b b minus c minus a 2 b again R3 same 2 c 2 c c minus a minus b Again by applying c1 goes to c1 minus c3 We get our left-hand side is equal to a plus b plus c Now c1 goes to c1 minus c3. So 1 minus 1 is 0 2 b minus 2 b is 0 2 c minus c minus a Minus b that is it is a plus b plus c So c2 and c3 are same 1 b minus c minus a 2 c 1 2 b c minus c minus b by taking out factor a plus b plus c Common from C1 we get Because on right-hand side in the question we want a plus b plus c2. So we are trying to bring a plus b plus c q our left-hand side is equal to a plus b plus c Into a plus b plus c So in column 1 we get 0 0 1 Column 2 and column 3 are same e minus c minus a 2 c 1 2 b C minus a minus b Now it is clear that we have to expand this determinant along c1 So on expanding along c1 we get our left-hand side is equal to a plus b plus c whole scale Into 1 into 1 b minus c minus a 1 2 b Which is equal to a plus b plus c whole square Into 1 into 2 b Minus 1 into b minus c minus a that is minus of b minus c minus a which is equal to a Plus b plus c whole square into 2 b minus b plus c plus a Which is again equal to a Plus b plus c whole square into a plus b plus c So we get this is equal to a Plus b plus c whole q which is equal to our right-hand side Hence we have proved that left-hand side is equal to right-hand side I hope the question is clear to you Bye and have a nice day