 Hello, I am welcome to the session. I am Deepika and I am going to help you to solve the following question. The question says Using properties of determinants prove the following determinant of A minus B minus C to A to A to B B minus C minus A to B 2C 2C C minus A minus B is equal to A plus B plus C whole cube So let's start the solution Let Delta is equal to given determinant whose elements are A minus B minus C to A to A to B B minus C minus A to B 2C 2C C minus A minus B So by using properties of determinant, we will prove that Delta is equal to A plus B plus C whole cube now by applying R1 goes to R1 plus R2 plus R3 we get Delta is equal to determinant of now the first element that is A11 of this determinant will become A minus B minus C plus 2B plus 2C which is equal to A plus B plus C similarly the element A12 is equal to 2A plus B minus C minus A plus 2C which is again equal to A plus B plus C again the element A13 is equal to 2A plus 2B plus C minus A minus B and this is again equal to A plus B plus C now the elements of R2 and R3 will remain the same now the elements of R2 are 2B B minus C minus A to B and the elements of R3 are 2C 2C C minus A minus B now here we observe that each element of R1 is A plus B plus C so by taking factor A plus B plus C common from R1 we get Delta is equal to A plus B plus C into determinant of 1 1 1 2B B minus C minus A 2B 2C 2C C minus A minus B now again by applying C2 goes to C2 minus C1 and C3 goes to C3 minus C1 we get Delta is equal to A plus B plus C into determinant of now C remains the same so the elements of first column are 1 2B and 2C now C2 goes to C2 minus C1 so C2 is equal to 1 minus 1 0 B minus C minus A minus 2B which is equal to minus of A plus B plus C again 2C minus 2C is equal to 0 now C3 goes to C3 minus C1 so 1 minus 1 is again equal to 0 2B minus 2B is equal to 0 C minus A minus B minus 2C is equal to minus of A plus B plus C now by expanding along R1 we get Delta is equal to A plus B plus C into 1 into A plus B plus C whole square minus 0 which is equal to A plus B plus C whole square again the elements A12 and A13 are 0 so we have Delta is equal to A plus B plus C whole cube but this is our right hand side hence our left hand side is equal to right hand side that is our given determinant is equal to A plus B plus C whole cube so this completes our session I hope the solution is clear to you bye and have a nice day