# Finding All Matrices Which Commute With a Given Matrix.

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 2x2 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!

• Try substitution and see if they agree. – Henricus V. Jan 12 '16 at 5:35

You've done everything just fine. You just need to perform the transformation $(a,b) \to (t,\frac{-s}{5}-t)$.
• Oh wow, I feel stupid now. Let a=t, $b=\frac{-s}{5}$ then s+5t=-5b and -s-4t=5b+a, thats so simple! One quick question though, is there any reason for doing this, or any way I should know that this is required? Thank you so much by the way. – CoffeeCrow Jan 12 '16 at 5:44