Show that`|[1+a,1,1],[1,1+b,1],[1,1,1+c]|=abc(1+1/a+1/b+1/c)=abc+bc+ca+ab`

Explanation

To show that

\[
\left| \begin{array}{ccc}
1 + a & 1 & 1 \\
1 & 1 + b & 1 \\
1 & 1 & 1 + c
\end{array} \right| = abc \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) = abc + bc + ca + ab,
...