# Thread: Find all matrices given constraint.

1. ## Find all matrices given constraint.

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

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

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 :

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

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

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

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

$AB=BA \Rightarrow ABC=BAC \Rightarrow AC=BAC \Rightarrow A=BA$ (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.