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QUESTION The set of all $2\times 2$ matrices with real entries
forms a vector space over \textbf{r}. Which of the following
subsets of matrices are subspaces?

\begin{description}

\item[(a)]
the set of those with zero trace;

\item[(b)]
the set of those with zero determinant;

\item[(c)]
the set of those with integer entries;

\item[(d)]
the set of symmetric matrices.

\end{description}


ANSWER
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\item[(a)]
Yes - if tr$A=$tr$B=0$ then tr$(A+B)=$tr$A$+tr$B=0$

and tr$(\lambda A)=\lambda$tr$A=0$.

\item[(b)]
No - it is easy to construct examples where $\det A=\det B=0$ but
$\det(A+B)\neq 0$.

\item[(c)]
No - if $A$ contains any odd numbers then $\frac{1}{2}A$ is not in
the set.

\item[(d)]
Yes - $A$ and $B$ symmetric imply both $A+B$ and $\lambda A$
symmetric.

\end{description}


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