Answer
The points $\left( 3,-1 \right),\left( 0,-3 \right)\text{ and }\left( 12,5 \right)\text{ are collinear}\text{.}$
Work Step by Step
The points $\left( {{x}_{1}},{{y}_{1}} \right),\left( {{x}_{2}},{{y}_{2}} \right)\text{ and }\left( {{x}_{3}},{{y}_{3}} \right)$ are collinear if,
$\left| \begin{matrix}
{{x}_{1}} & {{y}_{1}} & 1 \\
{{x}_{2}} & {{y}_{2}} & 1 \\
{{x}_{3}} & {{y}_{3}} & 1 \\
\end{matrix} \right|=0$
In order to check if the points $\left( 3,-1 \right),\left( 0,-3 \right)\text{ and }\left( 12,5 \right)$ are collinear, find out the determinant below:
$\left| \begin{matrix}
3 & -1 & 1 \\
0 & -3 & 1 \\
12 & 5 & 1 \\
\end{matrix} \right|$
Evaluate the above determinant by expanding along the first column as below:
$\begin{align}
& \left| \begin{matrix}
3 & -1 & 1 \\
0 & -3 & 1 \\
12 & 5 & 1 \\
\end{matrix} \right|=3\left| \begin{matrix}
-3 & 1 \\
5 & 1 \\
\end{matrix} \right|+12\left| \begin{matrix}
-1 & 1 \\
-3 & 1 \\
\end{matrix} \right| \\
& =3\left( -3-5 \right)+12\left( -1+3 \right) \\
& =3\left( -8 \right)+12\left( 2 \right) \\
& =0
\end{align}$
As this determinant is equal to zero, therefore, the provided points are collinear.
Thus, the points $\left( 3,-1 \right),\left( 0,-3 \right)\text{ and }\left( 12,5 \right)\text{ are collinear}\text{.}$
