If `a ,b` and `c` are all non-zero and `|[1+a,1, 1],[ 1,1+b,1],[1,1,1+c]|=0,` then prove that `1/a+1/b+1/c+1=0`

Explanation

To prove that \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 1 = 0 \) given that the determinant

\[
\begin{vmatrix}
1 + a & 1 & 1 \\
1 & 1 + b & 1 \\
1 & 1 & 1 + c
\end{vmatrix}
...