If vmatrix a-b & p-q & x-y b-c & q-r & y-z c-a & r-p & z-x vmatrix =k vmatrix a & b & c p & q & r x & y & z vmatrix , then what is the value of k ?

Step-by-step solution

Let = vmatrix a-b & p-q & x-y b-c & q-r & y-z c-a & r-p & z-x vmatrix Applying the row operation R₁ R₁ + R₂ + R₃ , we get: = vmatrix (a-b)+(b-c)+(c-a) & (p-q)+(q-r)+(r-p) & (x-y)+(y-z)+(z-x) b-c & q-r & y-z c-a & r-p & z-x vmatrix = vmatrix 0 & 0 & 0 b-c & q-r & y-z c-a & r-p & z-x vmatrix Since all elements of the first row are zero, the value of the determinant is 0 . Comparing this with k vmatrix a & b & c p & q & r x & y & z vmatrix , we get k = 0 .