 Hi and welcome to the session, I am Deepika here. Let's discuss the question by using properties of determinants, show that determinant a square plus 1 a v a c, a v b square plus 1 v c, c a c v c square plus 1 is equal to 1 plus a square plus b square plus c square. Let's start the solution. We have on the left hand side a square plus 1 a v a c, a v b square plus 1 v c, c a c v c square plus 1. By applying c 1 goes to a c 1, c 2 goes to b c 2 and c 3 goes to c c 3. We get our left hand side is equal to 1 over a b c into a into a square plus 1 a square v a square plus 1. c a square a b square b into b square plus 1 c b square a c square b c square and c into c q c square plus 1. Because we know that if delta 1 is a determinant obtained by applying c i to k c i to the determinant delta then delta 1 is equal to k delta or delta is equal to 1 by k delta 1. So, we get our left hand side is equal to 1 over a b c into a into a square plus 1 a square v c a square a b square b into b square plus 1 c b square a c square b c square c into c square plus 1. I hope this point is clear to you now. We can take common a b c from r 1, r 2, r 3 respectively. So, by taking out a b c common from r 1, r 2, r 3 respectively we get our left hand side is equal to a b c upon a b c into in r 1 we have taken out a. So, this is a square plus 1 b square c square and from r 2 we have taken b as a common factor. So, this is a square b square plus 1 c square and r 3 is our a square b square c square plus 1. Now, on cancellation we get our left hand side is equal to a square plus 1 b square c square a square b square plus 1 c square a square b square c square plus 1. By applying c 1 goes to c 1 plus c 2 plus c 3 we get our given determinant is equal to or left hand side is equal to a square plus 1 plus b square plus c square that is 1 plus a square plus b square plus c square 1 plus a square plus b square plus c square. Again we get same 1 plus a square plus b square plus c square and c 2 is as it is b square b square plus 1 b square c square c square and c square plus 1. Clearly we can take 1 plus a square plus b square plus c square common factor from c 1. So, we get this is equal to 1 plus a square plus b square plus c square 1 1 1 b square b square plus 1 b square c square c square c square plus 1. By applying r 1 goes to r 1 minus r 2 we get our left hand side is equal to 1 plus a square plus b square plus c square. Now r 1 goes to r 1 minus r 2 so 1 minus 1 0 b square plus 1 minus b square this is 1 minus 1 c square minus c square this is equal to 0. Again r 2 is as it is 1 b square plus 1 c square 1 b square c square plus 1. This is equal to 1 plus a square plus b square plus c square into minus of minus 1 into 1 c square 1 c square plus 1 by expanding along r 1. This is equal to 1 plus a square plus b square plus c square into c square plus 1 minus c square this is equal to 1 plus a square plus b square plus c square. In the question we have to prove our left hand side is equal to 1 plus a square plus b square plus c square so it means this is our right hand side. Hence left hand side is equal to right hand side hence proved. I hope the question is clear to you. Bye and have a nice day.