# Thread: Find all matrices given constraint.

1. ## Find all matrices given constraint.

Let $\displaystyle B = \begin{bmatrix}1&0\\0&0\end{bmatrix}, ~C=\begin{bmatrix}0&1\\0&0\end{bmatrix}$.

Find all 2x2 matrices A such that $\displaystyle AB=BA$ and $\displaystyle AC=CA$.

My analytic solution to this problem was clearly wrong as I got the single zero-matrix as a result. Clearly if A is the identity then those conditions are satisfied, but I don't know the analytic way to solve it correctly.

2. Hello,

Well, the brute force approach :

$\displaystyle A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$

By performing $\displaystyle AB=BA$, we get that $\displaystyle c,b=0$
By performing $\displaystyle AC=CA$, we get that $\displaystyle c=0,a=d$

So A is in the form $\displaystyle \begin{pmatrix}a&0\\0&a\end{pmatrix}=aI_2$

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Other thoughts :
$\displaystyle BC=C$ and $\displaystyle CB=0_2$

$\displaystyle AB=BA \Rightarrow ABC=BAC \Rightarrow AC=BAC \Rightarrow A=BA$ (because C is not the zero matrix)

Maybe it can be done with more abstract algebra (commutative things... ), but I don't know much of it.

3. Thanks alot, that helps.