If $ 
 A=
 \left[ {\begin{array}{cc}
 a & b \\
 c & d \\
 \end{array} } \right]
$
such that $trace(A)=0$

So it has to be in the form $ 
 A=
 \left[ {\begin{array}{cc}
 a & b \\
 c & -a \\
 \end{array} } \right]
$

How do I find the dimension and basis of this without any further given information?