I know this should be really simple, but for some reason I can;t figure it out.
I need to find all matrices which commute with the following 2×2 matrix A:
$$ B=\begin{bmatrix} 1 & -1\\ 5 & -4 \end{bmatrix} $$
I've tried using the definition of two commuting matrices, AB=BA and a generic matrix A where $ A=\begin{bmatrix} a & b\\ c & d \end{bmatrix} $ to generate a system of linear equations which I then solve to obtain a, b, c, and d such that A and B commute as follows:
$$ AB = \begin{bmatrix} a+5b & -a-4b\\ c+5d & -c-4d \end{bmatrix}=BA=\begin{bmatrix} a-c & b-d\\ 5a-4c & 5b-4d \end{bmatrix}$$
This gives the equations:
$$ c+ 5b=0 $$
$$ 5b-d+a=0 $$
$$ 5c-5a+5d=0 $$
$$5b+c=0 $$
This can then be converted to a matrix and reduced to row echelon form:
$$ \begin{bmatrix}
1 & 5 & 0 & -1 & 0 \\
0 & 5 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{bmatrix} $$
Introducing a parameter s for the variable c and a parameter t for the variable d, it seems this system can be solved to give:
$ a=s+t $, $b =\frac {-s} {5} $, $ c=s $, and $ d=t $.
Substituting these values into AB or BA yields the following matrix which, by rights should represent all matrices which commute with c:
$$ \begin{bmatrix} t & \frac {-s} {5} -t \\ s+5t & -s-4t \end{bmatrix} $$
However, the given solution is:
$$ \begin{bmatrix} a & b \\ -5b & 5b+a \end{bmatrix} $$
Our solutions seem quite similar (except of course for different variable names), but I jsut can;t for the life of me figure out where I've gone wrong, any help would be greatly appreciated!
Best Answer
You've done everything just fine. You just need to perform the transformation $(a,b) \to (t,\frac{-s}{5}-t)$.