Determine whether the set $H$ of all matrices form a subspace of $M_{2 \times 2}$ Determine if the set $Z$ of all matricies form
$ \left[ \begin{array}{cc}
a &amp; b \\
0 &amp; d 
\end{array} \right] $
is a subspace of $M_{2 \times 2}$ (the set of all $2 \times 2$ matrices). 

% This is something I came up with. Can someone look at it and let me know any useful corrections/suggestions to the question please.

Answer:

Without specification as to the nature of $a,b$ and $d$, it is assumed that $a,b,d \in \mathbb{R}$ 

Hence, $H$ is determined to be a subspace of $M_{2 \times 2}$ because it is closed under scalar addition and scalar multiplication and contains the zero vector when $a=b=d=0$.
What you have done looks reasonable except for one tiny point.  When you say "because it is closed under scalar addition and scalar multiplication", you should delete the first (but not the second) scalar, as you are trying to show that adding two elements give a third member of the subspace.  

So you should end with 


  Hence, $H$ is determined to be a subspace of $M_{2 \times 2}$ because it is closed under  addition and scalar multiplication and contains the zero vector when $a=b=d=0$.
Let $A=
\left[ 
\begin{array}{cc}
a &amp; b \\
0 &amp; d 
\end{array} 
\right]$ and 
$B=
\left[ \begin{array}{cc}
x &amp; y \\
0 &amp; z 
\end{array} \right]$. 
Then $(AB)_{2,1}=dz$ and not $0$. It doesn't seem to be closed under multiplication on its own set.