Answer
$A$ is an invertible matrix.
(Th.8, (a) and (c))
Work Step by Step
$A=\left[\begin{array}{llll}
-1 & -3 & 0 & 1\\
3 & 5 & 8 & -3\\
-2 & -6 & 3 & 2\\
0 & -1 & 2 & 1
\end{array}\right]\left\{\begin{array}{l}
\times(-1).\\
+3R_{1}.\\
-2R_{1}.\\
.
\end{array}\right.$
$\sim\left[\begin{array}{llll}
-1 & -3 & 0 & 1\\
0 & -4 & 8 & 0\\
0 & 0 & 3 & 0\\
0 & -1 & 2 & 1
\end{array}\right]\left\{\begin{array}{l}
.\\
\leftrightarrow R_{4}.\\
.\\
.
\end{array}\right.$
$\sim\left[\begin{array}{rrrrr}
-1 & -3 & 0 & 1\\
0 & -1 & 2 & 1\\
0 & 0 & 3 & 0\\
0 & -4 & 8 & 0
\end{array}\right]\left\{\begin{array}{l}
.\\
.\\
.\\
-4R_{2}.
\end{array}\right.$
$\sim\left[\begin{array}{rrrrr}
-1 & -3 & 0 & 1\\
0 & -1 & 2 & 1\\
0 & 0 & 3 & 0\\
0 & 0 & 0 & -4
\end{array}\right]$
we see 4 pivot positions in a 4$\times$4 matrix.
So, looking at Th.8,
$\mathrm{c}.\quad A$ has $n$ pivot positions.
is valid. Then,
$\mathrm{a}. \quad A$ is an invertible matrix.
is valid as well
$A$ is an invertible matrix.
(Th.8, (a) and (c))