If | array ccc x+y+2 z & x & y z & y+z+2 x & y z & x & z+x+2 y array |=k(x+y+z)^3 , then the value of k is

Step-by-step solution

aligned & = | array ccc x+y+2 z & x & y z & y+z+2 x & y z & x & z+x+2 y array | & = | array ccc 2(x+y+z) & x & y 2(x+y+z) & y+z+2 x & y 2(x+y+z) & x & z+x+2 y array | aligned (using C₁ C₁+C₂+C₃ ) Take out 2(x+y+z) common from C₁ , we get aligned & =2(x+y+z) | array ccc 1 & x & y 1 & y+z+2 x & y 1 & x & z+x+2 y array | & =2(x+y+z) | array ccc 1 & x & y 0 & y+z+x & 0 0 & 0 & z+x+y array | aligned (using R₂ R₂-R₁, R₃ R₃-R₁ ) Take out (x+y+z) common from R₂ and R₃ , we get aligned =2(x+y+z)(x+y+z)(x+y & +z) & | array l