 Hi and welcome to the session when we pick up here. Let's discuss a question by using properties of determinants show that Determinant x plus y plus 2z x y z y plus z plus 2x y Z x z plus x plus 2 y is equal to twice x plus y plus z whole cube Let's start the solution Solution Let us take the left hand side our left hand side is equal to x plus y plus 2 z z z x y plus z plus 2 x x y y z plus x plus 2 y Clearly we can see that if we will operate C1 goes to C1 plus C2 plus C3 then We will able to solve this question by applying C1 goes to C1 plus C2 plus C3 We get our left hand side is equal to C1 Plus C2 plus C3 that is this is equal to 2x plus 2 y plus 2 z again z plus y plus z plus 2x plus 5 is Equal to 2x plus 2 y plus 2 z again z plus x plus Z plus x plus 2 y 2 x plus 2 y Plus 2 z now C2 and C3 are same now by taking out x plus y plus z common form C1 we get our left hand side is equal to twice x plus y plus z into 1 1 1 x y plus z plus 2 x x y y z plus x plus 2 y By applying 2 goes to R2 minus R1 and R3 goes to R3 minus R1 We get our left hand side is equal to twice x plus y plus z Now R2 goes to R2 minus R1. So R1 is same 1 x y and This is 1 minus 1 0 y plus z plus 2 x minus x is Y plus z plus x and y minus y is also 0 now R3 also goes to R3 minus R1 That is 1 minus 1 0 x minus x 0 z plus x plus 2 y minus y is z plus x plus y Now expanding along R3 we get our left hand side is equal to twice x plus y plus z into x plus y plus z into 1 into y plus z plus x minus 0 That is by expanding So we get this is equal to twice x plus y plus z whole square into x plus y plus z which is equal to Twice x plus y plus z whole cube. This is equal to our right hand side Hence our left hand side is equal to right hand side Hence proved. I hope the question is clear to you. Bye and have a nice day