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

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Solution

Augumented matrix is $⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 & 1 1 & 2 & 4 & p 1 & 4 & 10 & p^{2} \end{matrix} ⎤ ⎥ ⎦$

applying $R_{3} \to R_{3} - R_{1}$ and $R_{2} \to R_{2} - R_{1}$ gives

$\to ⎡ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 & 1 0 & 1 & 3 & p - 1 0 & 3 & 9 & p^{2} - 1 \end{matrix} ⎤ ⎥ ⎦$

dividing $R_{3}$ with $3$ gives

$\to ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 1 & 1 & 1 0 & 1 & 3 & p - 1 0 & 1 & 3 & \frac{p^{2} - 1}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$

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)

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

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