Question

The sum of values of $p$ for which the equations $x+y+z=1,x+2y+4z=p,x+4y+10z=p^2$ have a solution is ____

Answer 1

Augumented matrix is $\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& 2& 4& p\\ 1& 4& 10& p^2\end{array}\right]$

applying $R_3\to R_3-R_1$ and $R_2\to R_2-R_1$ gives

$\to \left[\begin{array}{cccc}1& 1& 1& 1\\ 0& 1& 3& p-1\\ 0& 3& 9& p^2-1\end{array}\right]$

dividing $R_3$ with $3$ gives

$\to \left[\begin{array}{cccc}1& 1& 1& 1\\ 0& 1& 3& p-1\\ 0& 1& 3& \frac{p^2-1}{3}\end{array}\right]$

Given system has a solution if the determinant of the above matrix is 0,

it is possible when $\frac{p^2-1}{3}=p-1$ (When any two rows of matrix are same, it's the determinant is zero)

$\Rightarrow p^2-3p+2=0$
$\Rightarrow p=1,2$
$\therefore$ Sum of values of $p$ $=3$