Analyzing The Matrix

$\large Given$ :

$\large det\begin{pmatrix} p & b & c\\ a & q & c\\ a & b & r\\ \end{pmatrix}=0$

$\large Now$ ,

$\large R_{1}$ ➝ $\large R_{1} - R_{3}$ , $\large R_{2}$ ➝ $\large R_{2} - R_{3}$

$\large det\begin{pmatrix} p-a & 0 & c-r\\ 0 & q-b & c-r\\ a & b & r\\ \end{pmatrix}=0$

$\large On$ $\large expanding$ ,

$\large (p-a)$ $\large ((q-b)r - b(c-r))$ $\large +$ $\large (c-r)(-a(q-b))$ $\large =$ $\large 0$

$\large (p-a)$ $\large ((q-b)r + b(r-c))$ $\large +$ $\large (r-c)a(q-b)$ $\large =$ $\large 0$

$\large On$ $\large dividing$ $\large by$ $\large (q-b)(r-c)$ $\large on$ $\large both$ $\large sides$ ,

$\large (p-a)$ ( $\large \frac{r}{r-c} + \frac{b}{b-q}$ ) $\large +$ $\large a$ $\large =$ $\large 0$

$\large On$ $\large dividing$ $\large by$ $\large p-a$ $\large on$ $\large both$ $\large sides$ ,

$\large \frac{r}{r-c} + \frac{b}{q-b} + \frac{a}{p-a}$ $\large =$ $\large 0$

$\large Adding$ $\large 2$ $\large on$ $\large both$ $\large sides$ ,

$\large \frac{r}{r-c} + \frac{b}{q-b} + 1 + \frac{a}{p-a} + 1$ $\large =$ $\large 2$

$\large \frac{r}{r-c} + \frac{q}{q-b} + \frac{p}{p-a}$ $\large =$ $\large 2$