Mathematics - Determinant Question with Solution | TestHub

Let $|A|=\left|\begin{array}{ccc}p+a& b& c\\ a& q+b& c\\ a& b& r+c\end{array}\right|$ Also, given that equations has non trivial solution $|A|=0$ $\begin{array}{rl}& \Rightarrow \left|\begin{array}{ccc}p+a& b& c\\ a& q+b& c\\ a& b& r+c\end{array}\right|=0\\ & \Rightarrow (p+a)[(q+b)(r+c)-bc]-b[a(r+c)-ca]\\ & +c[ab-a(q+b)]=0\\ & \Rightarrow (p+a)[qr+qc+br]-b[ar]+c[-aq]=0\\ & \Rightarrow pqr+pqc+pbr+aqr=0\end{array}$ Dividing whole equation by $pqr$ $\begin{array}{rlr}\Rightarrow & \frac{pqr}{pqr}+\frac{pqc}{pqr}+\frac{prb}{pqr}+\frac{qra}{pqr}& =0\\ \Rightarrow & 1+\frac{c}{r}+\frac{b}{q}+\frac{a}{p}& =0\\ \Rightarrow & \frac{a}{p}+\frac{b}{q}+\frac{c}{r}& =-1\end{array}$