Question

Obtain the value of \[ \Delta= \begin{vmatrix} 1+x & 1 & 1\\ 1 & 1+y & 1\\ 1 & 1 & 1+z \end{vmatrix} \] in terms of \(x\), \(y\) and \(z\).

Further, if \[ \Delta=0 \] and \(x\), \(y\) and \(z\) are non-zero real numbers, prove that \[ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=-1. \]

Hint:

Factoring out variables to create fraction forms like \(\frac{1}{x}\) is a useful technique for solving determinants that contain cyclic variables on their main diagonal.

Answer 1