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

$\text{L.H.S}$
$\text{Let}$ $\Delta =\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|Takecommonafrom{C}_1,bfrom{C}_2,cfrom{C}_3$

$=abc\left|\begin{array}{ccc}\frac{1}{a}+1& \frac{1}{b}& \frac{1}{c}\\ \frac{1}{a}& \frac{1}{b}+1& \frac{1}{c}\\ \frac{1}{a}& \frac{1}{b}& \frac{1}{c}+1\end{array}\right|C_1\to C_1+C_2+C_3$

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

$=abc(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\left|\begin{array}{ccc}1& 1/b& 1/c\\ 1& 1/b+1& 1/c\\ 1& 1/b& 1/c+1\end{array}\right|R_2\to R_2-R_1{R}_3\to R_3-R_1$

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

$=abc(1+\frac{1}{a}+\frac{1}{b}+\frac{1}{c}).1=abc+bc+ca+ab$