Answer
It is not a vector space.
Work Step by Step
The set of all $2\times 2$ nonsingular matrices with the
standard operations is not a vector space since it is not closed under addition. For example, the matrices
$$
\left[\begin{array}{lll}{2} & {1} \\ {0} & {1}\end{array}\right], \quad \left[\begin{array}{lll}{-2} & {1} \\ {0} & {1}\end{array}\right]
$$
are nonsingular but thier sum is $\left[\begin{array}{lll}{0} & {1} \\ {0} & {1}\end{array}\right]$
which is singular.
