Explanation

(c)
$\begin{bmatrix}
1 & x & y
\end{bmatrix}\begin{bmatrix}
1 & 3 & 1 \\
0 & 2 & - 1 \\
0 & 0 & 1
\end{bmatrix} = \begin{bmatrix}
1 \\
x \\
y
\end{bmatrix} = 10$
$\Rightarrow \lbrack 1\ \ \ 3 + 2x\ \ \ 1 - x + y\rbrack\begin{bmatrix}
1 \\
x \\
y
\end{bmatrix} = \lbrack 0\rbrack$
$\Rightarrow \left\lbrack 1 + 3x + 2x^{2} + y - xy + y^{2} \right\rbrack = \lbrack 0\rbrack$
$\Rightarrow 2x^{2} + y^{2} + y + 3x - xy + 1 = 0$
If $y = 0$,$2x^{2} + 3x + 1 = 0$
$\Rightarrow (2x - 1)(x + 1) = 0$
$\Rightarrow x = - \frac{1}{2},\ - 1$(rational roots)
If $y = - 1,\ 2x^{2} + 4x + 1 = 0$
$\Rightarrow x = \frac{- 4 \pm \sqrt{12}}{4} = \frac{- 2 \pm \sqrt{3}}{2}$ (irrational roots)