Answer
It is not a vector space.
Work Step by Step
The set of all $2x2 $ matrices of the form $$
\left[\begin{array}{ll}{a} & {b} \\ {c} & {1}\end{array}\right]
$$
with standard operations is not a vector space. For example, it is not closed under addition,
$$\left[\begin{array}{ll}{1} & {0} \\ {1} & {1}\end{array}\right]
+\left[\begin{array}{ll}{2} & {1} \\ {0} & {1}\end{array}\right]
=\left[\begin{array}{ll}{3} & {1} \\ {1} & {2}\end{array}\right].
$$ which is not an element of the set.