Mathematics - Determinant Question with Solution | TestHub

Since the system is homogeneous and has a non-trivial solution $0=\left|\begin{array}{ccc}p+a& b& c\\ a& q+b& c\\ a& b& r+c\end{array}\right|=\left|\begin{array}{ccc}p& -q& 0\\ 0& q& -r\\ a& b& r+c\end{array}\right|$ using ${\mathrm{R}}_1\to {\mathrm{R}}_1-{\mathrm{R}}_2$ and ${\mathrm{R}}_2\to {\mathrm{R}}_2-{\mathrm{R}}_3$ Expanding along $\mathrm{C}$, we get $\Rightarrow \mathrm{p}[\mathrm{q}(\mathrm{r}+\mathrm{c})+\mathrm{br}]+\mathrm{aqr}=0$ $\Rightarrow \mathrm{pqr}+\mathrm{pqc}+\mathrm{prb}+\mathrm{qra}=0$ $\Rightarrow \frac{a}{p}+\frac{b}{q}+\frac{c}{r}=-1$