 Hi, I'm Meenu and going to help you to solve the following problem. It says prove the following using properties of determinants So let's move on to the solution By the properties of determinants. We know that if we Apply any row operation or column operation to the determinant its value doesn't change it remains the same So we can apply any row operation or column operation to simplify the given determinant Now looking at the expression on the right hand side of the question We can see that we want to have a plus b plus c so we apply the column operation as C1 is equal to C1 plus C2 plus C3 and We can see that after applying this column operation the first column becomes a plus b plus c and the second and the Third column remains as it is Now we take a plus b plus c common from the first column minus a plus c minus b plus c 3c again, we apply the row operation R2 is equal to R2 minus R1 And R3 is equal to R3 minus R1 1 minus a plus b minus a plus c 0 3b minus minus a plus b and Minus b plus c minus a plus c 0 minus c plus b minus minus a plus b then 3c minus minus a plus c again, this is equal to 1 minus a plus b minus a plus c 0 a plus 2b minus b and we have 0 Plus b minus b gets cancelled and we have a minus c and here we have a Plus 2c Now we expand this determinant along first row so that we get 1 into a plus 2c into a plus 2b minus a minus c into a minus b and the other terms would be 0 and this is again equal to a plus b plus c into Open the brackets a square plus 2ab plus 2ac Plus 4bc minus a square minus minus a b But when we open the bracket it will become plus a b Then we have Minus ac But it becomes plus a b when we open the bracket then we have bc which becomes minus bc and We can see that a square gets cancelled with minus a square and we get a plus b plus c into 2ab plus a b is 3ab 2ac plus ac is 3ac 4bc minus bc is 3bc So this becomes equal to 3 into a plus b plus c into a b plus bc Plus ca and this is what we had to prove so hence the result is proved So this completes the question. Bye for now. Take care. Hope you enjoy the session