For what real values of $k$ , the system of equation $x + 2 y + z = 1; x + 3 y + 4 z = k; x + 5 y + 10 z = k^{2}$ has solution? Find the sum of all the possible values of k?

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Solution

Augumented matrix is $⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 1 & 1 1 & 3 & 4 & k 1 & 5 & 10 & k^{2} \end{matrix} ⎤ ⎥ ⎦$
applying $R_{3} \to R_{3} - R_{1}$ and $R_{2} \to R_{2} - R_{1}$ gives
$\to ⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 1 & 1 0 & 1 & 3 & k - 1 0 & 3 & 9 & k^{2} - 1 \end{matrix} ⎤ ⎥ ⎦$
dividing $R_{3}$ with $3$ gives
$\to ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 2 & 1 & 1 0 & 1 & 3 & k - 1 0 & 1 & 3 & \frac{k^{2} - 1}{3} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦$
given system has solution if $\frac{k^{2} - 1}{3} = k - 1$
$\Rightarrow k^{2} - 3 k + 2 = 0$
$\Rightarrow k = 1, 2$
$∴$ Sum of values of $k$ $= 3$

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