Question

If $a,b$ and $c$ are all non-zero and $\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|=$0, then prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+$1$=$0.

Answer 1

We have,

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

$$\begin{gathered} C_1\to C_1-C_2 \\ \left|\begin{array}{ccc}a& 1& 1\\ -b& 1+b& 1\\ 0& 1& 1+c\end{array}\right|=0 \\ {C}_2\to C_2-C_3 \\ \left|\begin{array}{ccc}a& 0& 1\\ -b& b& 1\\ 0& -c& 1+c\end{array}\right|=0 \\ \mathrm{Expanding}\mathrm{along}{R}_1,\mathrm{we}\mathrm{get} \\ a(b+bc+c)+1\left(bc\right)=0 \\ \Rightarrow ab+abc+ac+bc=0 \\ \mathrm{Dividing}\mathrm{by}abc,\mathrm{we}\mathrm{get} \\ \frac{1}{c}+1+\frac{1}{b}+\frac{1}{a}=0 \\ \therefore \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1=0 \end{gathered}$$