How to approach this question on subspaces? 
Determine which of the following are subspaces of $3 \times 3$ matrix $M$
  all $3 \times 3$ matrices $A$ such that the trace of $A$ is $\mbox{tr}(A) = 0$.

What does trace mean?

Let $V$ be a vector space.
Then $W \subset V$ is a subspace of $V$ if
a) $0 \in W$;
b) $u, v \in W$ then $u + v \in W$;
c) $u \in W$ e $\lambda \in \mathbb{R}$, then $\lambda u \in W$.
The trace of a matrix $A$ of order 3 is given by $\mbox{tr} (A) = a_{11} + a_{22} + a_{33}$ (it is the sum of the principal diagonal).
Then the set $W$ formed by all matrices of order 3 x 3 whose trace is zero is a subspace of matrices of order 3.
In fact, just show the 3 items above.
a) the null matrix is in $W$ (note that the trace is zero).
b) Let $A, B \in W$, then $\mbox{tr} (A) = \mbox{tr} (B) = 0$. Hence $\mbox{tr} (A + B) = \mbox{tr} (A) + \mbox{tr} (B) = 0 + 0 = 0$.
c) Let $A \in W$ and $\lambda \in \mathbb{R}$, then $\mbox{tr} (\lambda A) = \lambda a_{11} + \lambda a_{22} + \lambda a_{33} = \lambda (a_{11} + a_{22} + a_{33}) = \lambda \mbox{tr} (A)$.
And the result is proven.
