 Hello and welcome to the session. Let's discuss the following question. It says using the properties of determinant show that the determinant a plus b, b plus c, c plus a, b plus c, c plus a, a plus b, c plus a, a plus b, b plus c is equal to the twice of the determinant a, b, c, b, c, a, c, a b. So, let us now start the solution and consider the determinant a plus b, b plus c, c plus a, b plus c, c plus a, a plus b, c plus a, a plus b, b plus c. We have to prove that this determinant is equal to this determinant. So, to prove this, we'll apply any row or column operation. So, here we'll apply the column operation. The c1 is equal to c1 plus c2 minus c3. So, we have c1 plus c2 minus c3 that is 2b, 2c, 2a, a plus b plus b plus c minus. c minus a gives us 2b, b plus c plus c plus a minus a minus b gives us 2c and c plus a plus a plus b minus b minus c gives us 2a and second and the third column remains as it is. So, we have b plus c, c plus a, a plus b, c plus a, a plus b, b plus c. Now, taking two common from the first column we have, b, c, a is the first column, b plus c, c plus a, a plus b, c plus a, a plus b, b plus c. Now, again, we'll apply the column operation. So, c2 equal to c2 minus c1. So, we have the determinant twice of b, c, a. First column remains as it is and second column is c2 minus c1 that is b plus c minus b is c, c plus a minus c is a, a plus b minus a is b and the third column remains as it is. Again, we apply the column operation as c3 equal to c3 minus c2. So, we have twice of the determinant b, c, a, c, a, b. Now, we have column operation c3 minus c2 that is c plus a minus c is a, a plus b minus a is b, b plus c minus b is c. Now, looking at what we have to prove, a, b, c should be the first column, b, c, a should be the second column and c, a, b should be the third column. So, we'll interchange the columns. So, a, b, c becomes the first column if we replace it by third one, right? So, c1 becomes c3 and c3 becomes c1. So, we have determinant a, b twice of a, b, c, c, a, b remains the second one only right now and this becomes the third one b, c, a and since we have interchanged the column, we have negative sign outside. Again, we need to have b, c, a, a as the second column and c, a, b as the third column. So, we'll interchange c2 by c3 and c3 by c2. So, c2 becomes c3 and c3 becomes c2. And again, since we are interchanging, we'll have negative sign but negative, negative becomes positive. So, we have twice of the determinant a, b, c. The second column becomes the third and third becomes the second. So, second is b, c, a, third is c, a, b. And this is what we had to prove. So, this completes the question and the session. Bye for now. Take care. Have a good day.