 Hi and welcome to the session. I am Deepika here. Let's discuss a question. The question says, using the property of determinants and without expanding, prove that determinant v plus cq plus ry plus z, c plus ar plus p z plus x, a plus vp plus q x plus y, this determinant is equal to twice determinant apxvqy, c, r, z. Let's start the solution. Take the left hand side of our determinant and left hand side is b plus c, c plus a, a plus p, q plus r, r plus p, p plus q, and y plus z, z plus x, x plus y. We will prove that left hand side is equal to right hand side. By applying to r1 plus r2 plus r3, we get our left hand side is equal to b plus c, c plus q plus a plus b, and again we have q plus r plus r plus p plus p plus q. Again we have y plus z plus z plus x plus x plus y, r2 and r3 are same, c plus q, r plus p, z plus x, a plus b, p plus q, and x plus y. This is equal to twice a plus b plus c. In the first row, we get again twice p plus q plus r, twice x plus y plus z. 2 is same, 3 is also same. Again we can rewrite this as twice a plus b plus c, p plus q plus r, x plus y plus z, c plus q, r plus p, z plus x, a plus b, p plus q, x plus y. By using the property, if each element of a row or a column of a determinant is multiplied by a constant k, then its value gets multiplied by k. By applying, r2 goes to r2 minus r1 and r3 goes to r3 minus r1. We get our left hand side is equal to twice, r1 is same, a plus b plus c, p plus q plus r, and x plus y plus z. R2 minus r1 that is a plus c minus a minus b minus c, r plus p minus p minus q minus r, and z plus x minus x minus y minus z. And r3 is r3 minus r1 that is a plus b minus a minus b minus c, p plus q minus p minus q minus r, and x plus y minus x minus y minus z. On cancellation, we get, in r2 we get, so our left hand side is equal to twice, a plus b plus c, r1 is same, p plus q plus r, x plus y plus z. And this is equal to minus b minus q minus y and here in r3 we get minus c minus r minus z. Again by applying, r1 goes to r1 plus r2 plus r3, we get left hand side is equal to twice, a, because when we will add a plus b plus c minus b minus c, we will get, on cancellation we will get only a. Here we will get only p, and here we will get x, because minus y and minus z will be cancelled by y and z. Again r2 and r3 are same. Again we can rewrite this as, this is equal to 2 into minus 1 into minus 1, a, p, x, b, q, y, c, r, z by using the above property, the above property. Now this is equal to twice p, x, b, q, y, c, r, z, which is equal to right hand side, because on right hand side we have given this only. Hence we have proved that left hand side is equal to right hand side. I hope the question is clear to you, why and have a good day.