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 solve the problem, we need to prove that if the determinant of the matrix

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