<s>[INST] How do I find the centralizer for this matrix? 
Let $B=\begin{pmatrix}
2 & 2 \\
 0& 1 \\
\end{pmatrix} \in GL_2(\mathbb{R})$. Determine the centralizer for $B$.

Here's what I've done so far:
Let $D=\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix} \in GL_2(\mathbb{R})$.
$\begin{align} 
BD&=DB\\
\begin{pmatrix}
2 & 2 \\
 0& 1 \\
\end{pmatrix} \begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix} &= \begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix} \begin{pmatrix}
2 & 2 \\
 0& 1 \\
\end{pmatrix}\\
\begin{pmatrix}
2(a+c) & 2(b+d) \\
 c& d \\
\end{pmatrix}&= \begin{pmatrix}
2a & 2a+b \\
 2c& 2c+d \\
\end{pmatrix}
\end{align} $
From the above, we have:
$\begin{aligned}
2a+2c&=2a\\
2b+2d&=2a+b\\
c&=2c\\
d&=2c+d
\end{aligned}
$
From which we derive:
$\begin{aligned}
c&=0\\
b+2d-2a&=0\\
d&=d \in \mathbb{R} \backslash \{0\}
\end{aligned}$
As you can see, I've managed to determine $c$ and $d$, but I can't seem to figure out $a$ and $b$. What am I missing?
 [/INST] This means that an invertible $2 \times 2$ matrix $D$
commutes with $B$ if and only if it is of the form
$$D=\begin{pmatrix}
a & 2a-2d \\
0 & d \\
\end{pmatrix}$$
for $a,d \in \mathbb{R}\setminus\{0\}$.
</s>