Finding a basis for a set of matrices in a vector space. I am trying to find a basis for the following vector space: 
V = {2x2 matrices A | $\bigl( \begin{smallmatrix} 1 & 2 \\ 2 & 3 \end{smallmatrix} \bigr)$A = A$\bigl( \begin{smallmatrix} 1 & 2 \\ 2 & 3 \end{smallmatrix} \bigr)$}
So far, I have augmented $\bigl( \begin{smallmatrix} 1 & 2 \\ 2 & 3 \end{smallmatrix} \bigr)$ with the identity matrix to find that A = $\bigl( \begin{smallmatrix} -3 & 2 \\ 2 & -1 \end{smallmatrix} \bigr)$. Is the identity matrix the only basis for this question? 

Start with the definitions
$$
  \mathbf{A} = 
\left[
\begin{array}{cc}
 1 & 2 \\
 2 & 3 \\
\end{array}
\right],
\qquad
  \mathbf{B} = 
\left[
\begin{array}{cc}
 a & b \\
 c & d \\
\end{array}
\right].
$$
Following the logic of @DonAntonio, compute the commutator:
$$
 \left[ \mathbf{A}, \mathbf{B} \right] = 
   \mathbf{A} \mathbf{B} - 
   \mathbf{B} \mathbf{A} = 
   2\left[ \begin{array}{cc}
      -b + c & -a - b + d \\
     a+c-d  & b -c
    \end{array} \right]
= \mathbf{0}.
$$
From the diagonal terms we see that $b = c$. We now have matrix.
$$
  \mathbf{B} = 
\left[
\begin{array}{cc}
 a & b \\
 b & c \\
\end{array}
\right], \qquad
\left[ \mathbf{A}, \mathbf{B} \right] = 
   2\left[ \begin{array}{cc}
      0 & -a - b + d \\
     a+b-d  & 0
    \end{array} \right]
= \mathbf{0}.
$$
From this, $d = a+b$. 
The set of matrices which commute with $\mathbf{A}$ are
$$
\mathbf{B} = 
\left[
\begin{array}{cc}
 a & b \\
 b & a + b \\
\end{array}
\right], \quad \alpha, \beta \in \mathbb{C}.
$$
Notice the original matrix $\mathbf{A}$ has this form.
