 Hello and welcome to the session. I am Deepika here. Let's discuss the question by using properties of determinants show that Determine 1 x x square x square 1 x x x square 1 is equal to 1 minus x q whole square Let us start the solution Solution let us take the left hand side our left hand side is equal to 1 x x square x square 1 x x square 1 by applying r1 goes to r1 plus r2 plus r3 We get our left hand side is equal to 1 plus x square plus x x plus 1 plus x square x square plus x plus 1 R2 is as it is R3 is also as it is Now we can see that we can take 1 plus x plus x square Common in row 1 so by taking out 1 plus x plus x square Common From r1 we get our left hand side is equal to 1 plus x plus x square into 1 1 1 x square 1 x x x square 1 now by applying 2 goes to r2 minus r1 and R3 goes to r3 minus r1 We get our left hand side is equal to 1 plus x plus x square into 1 1 1 Now r2 goes to r2 minus r1 that is x square minus 1 1 minus 1 0 x minus 1 R3 also goes to r3 minus r1 so x minus 1 x square minus 1 1 minus 1 0 clearly x minus 1 is Common in r2 as well as in r3 now by taking out factor x minus 1 common from R2 and r3 We get our left hand side is equal to 1 plus x plus x square into x minus 1 square into 1 1 1 Now from r2 we have taken out common factor x minus 1 so here we get x plus 1 0 1 and r3 also we have taken out x minus 1 so 1 x plus 1 0 this is equal to 1 plus x plus x square into x minus 1 whole square 1 into 1 0 1 1 minus x plus 1 into 1 1 x plus 1 1 this we get get When we expand along r3 This is again equal to this is equal to 1 plus x plus x square into x minus 1 whole square into 1 into 1 minus 0 minus x plus 1 into 1 minus x minus 1 So here we can cut 1 with minus 1 We are left with 1 plus x plus x square into x minus 1 whole square 1 plus x Square plus x This is again equal to which is equal to x minus 1 whole square into 1 plus x square plus x whole square This we can rewrite as 1 minus x into 1 plus x square plus x whole square This is equal to now we know the formula that a minus b into a Square plus a b plus b square is equal to a q minus b q So here we get 1 minus x q Whole scale Because here are a is equal to 1 and b is equal to x. So this is our right-hand side This is equal to our right-hand side. Hence we have to hence Left-hand side is equal to right-hand side Hence prove I Hope the question is clear to you my and have a good day