 Hi and welcome to the session. I am Deepika here. Let's discuss a question. Question says, by using properties of determinants, show that determinant 1a a square, 1b b square, 1cc square is equal to a minus v into b minus c into c minus a. Let's start the solution. Solution. Let us take the left hand side. Our left hand side is equal to 1, 1, 1, a, b, c, a square, b square, c square. We will prove that our left hand side is equal to right hand side. Now, by applying r1 goes to r1 minus r2 and r2 goes to r2 minus r3, we get our left hand side is equal to r1 goes to r1 minus r2 that is 1 minus 1, 0, a minus b and a square minus b square. And r2 goes to r2 minus r3 that is 1 minus 1, 0, b minus c, b square minus c square. And r3 is same, 1cc, 1cc square. We can see that in this determinant we can take a minus b common from the r1 and b minus c from r2. So, we get this is equal to a minus b, b minus c is equal to 0, 1, a plus b, 0, 1, b plus c, 1cc square on taking a minus b and b minus c common from r1 and r2. By applying r1 goes to r1 minus r2, we get our left hand side is equal to a minus b, b minus c, 0, 0, a minus c, 0, 1, b plus c, 1cc square. Now, expanding along r1 we get our left hand side is equal to a minus b, b minus c, 0, minus 0, plus a minus c and 0, 1, 1c that is by expanding along r1. This is equal to a minus b, b minus c, a minus c into 0, minus 1 that is minus 1 which is again equal to a minus b, b minus c, c minus a which is equal to r right hand side. Hence, left hand side is equal to right hand side. I hope the question is clear to you. Bye and have a nice day.