 Hi, and welcome to the session. I am Deepika here. Let's discuss the question by using properties of determinants Show that determinant x x square y z y y square z x z z square x y is equal to x minus y into y minus z into z minus x into x y plus y z plus z x So let's start the solution We have on the left hand side y z y y square z x z z square x y goes to r2 minus r1 3 goes to R1 then side is equal to this as it is x x square y z r2 goes to r2 minus r1. So this is y minus x y square minus x square z x minus y z 3 goes to r3 minus r1. So we have this is z minus x z square minus x square and this is x y minus y z. This is again equal to y minus x z minus x x square y plus x into y minus x z plus x into z minus x again, this is y z and this is z y and this is y into x minus y minus x is common in r2 and z minus x is common in r3 So by taking out our left hand side is equal to y z. Now y minus x is common here. So this is 1 y plus x and this is minus z. Again z minus x is common from r3 So we are left with 1 z plus x and this is minus y goes to r3 minus r2 We get that is equal to This is equal to y minus x into z minus x y z 1 y plus x 3 goes to r3 minus r2. So this is 1 minus 1 0 this is z plus x minus y minus x. So this is z minus y and This is minus y plus z. Similarly z minus y is common in r3. So by taking out common y common from y minus x into z minus x into z minus y This is x x square y z 1 y plus x minus z is equal to minus x z minus x z minus y It goes to c2 minus c3. So this is x square minus y z y plus x plus z This is 1 minus 1 0. So again y z minus z and 1. We get This is equal to now y minus x is can be written as x minus y and And z minus y is y minus z into z minus x 3. So we get x into into x plus y plus z 1 into x square minus y so this is minus x square plus y z So this is equal to this y y minus z z minus x Into plus x y plus x z minus x square plus y z. So this is equal to x minus y y minus z z minus x into x y plus y z Plus z x so this is our right hand side. This only we have to prove So we have proved that our given determinant is equal to x minus y into y minus z into z minus x Into x y plus y z plus z x hence left hand side equal to right hand side Hence proved this question is clear to you. Why you enjoy yourself?