Question

Using properties of determinants, prove that:
$\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=abc+bc+ca+ab$

Answer 1

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

$R_1\to R_1-R_2{R}_3\to R_3-R_2$

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

$=a(c+bc+b)+bc$

$=ab+bc+ac+abc$