If a,b and c are all non-zero

We have,

$\left|\begin{array}{ccc}1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c\end{array}\right|=0$

$C_{1} \rightarrow C_{1}-C_{2}$

$\left|\begin{array}{ccc}a & 1 & 1 \\ -b & 1+b & 1 \\ 0 & 1 & 1+c\end{array}\right|=0$

$C_{2} \rightarrow C_{2}-C_{3}$

$\left|\begin{array}{ccc}a & 0 & 1 \\ -b & b & 1 \\ 0 & -c & 1+c\end{array}\right|=0$

Expanding along $R_{1}$, we get

$a(b+b c+c)+1(b c)=0$

$\Rightarrow a b+a b c+a c+b c=0$

Dividing by $a b c$, we get

$\frac{1}{c}+1+\frac{1}{b}+\frac{1}{a}=0$

$\therefore \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1=0$