Answer
See the counterexample below.
Work Step by Step
Consider the following statement:
The set of all $3\times 3$ matrices of the form
$$A=\left[\begin{array}{ccc} 0 & a & b\\ c & 2 & d\\ e & f & 0\end{array}\right]$$
with the standard operations is a vector space.
We will now present a counterexample in order to show that the statement is false.
Consider the matrix
$$A=\left[\begin{array}{ccc} 0 & 1 & 3\\ 5 & 2 & 7\\ 9 & 11 & 0\end{array}\right]$$
which has the form required above.
Observe that
$$3A=\left[\begin{array}{ccc} 0 & 3 & 9\\ 15 & 6 & 21\\ 27 & 33 & 0\end{array}\right]$$
which is not of the form required above.
Hence, the set of all matrices of the form
$$A=\left[\begin{array}{ccc} 0 & a & b\\ c & 2 & d\\ e & f & 0\end{array}\right]$$
with the standard operations is not closed under scalar multiplication and therefore, it is not a vector space.
