Answer
It is not a vector space.
Work Step by Step
The set of all $3\times3$ matrices of the form
$$
\left[\begin{array}{lll}{1} & {a} & {b} \\ {c} & {1} & {d} \\ {e} & {f} & {1}\end{array}\right]
$$
with the standard operations is nota vector space. One can see easilly that the set is notclosed under addition. For example,
$$\left[\begin{array}{lll}{1} & {0} & {1} \\ {1} & {1} & {2} \\ {2} & {0} & {1}\end{array}\right]+\left[\begin{array}{lll}{1} & {2} & {0} \\ {0} & {1} & {1} \\ {2} & {0} & {1}\end{array}\right]=\left[\begin{array}{lll}{2} & {2} & {1} \\ {1} & {2} & {3} \\ {4} & {0} & {2}\end{array}\right]$$
which is not an element of the set.
