Prove Commutation Property for 2x2 Matrices D

[vdgreat]

Let D =

[d11 d12]
[d21 d22]

be a 2x2 matrix. Prove that D commutes with all other 2x2
matrices if and only if d12 = d21 = 0 and d11 = d22.

I know if we can prove for every A, AD=DA should be true, but I really don't know how to proceed from there. I tried equating elements of AD with DA but that really didnt help.

Can anyone help me with this problem. Thanks..

 

[morphism]

Try particular As, like

$$A = \left(\begin{array}{cc} 0 & 1 \\ 0 & 0\end{array}\right),$$

and see where that leads you.

 

[HallsofIvy]

morphism's idea is excellent. Do you see where he got it?
The matrices
$$\left(\begin{array}{cc}1 & 0 \\ 0 & 0 \end{array}\right)$$
$$\left(\begin{array}{cc}0 & 1 \\ 0 & 0 \end{array}\right)$$
$$\left(\begin{array}{cc}0 & 0 \\ 1 & 0 \end{array}\right)$$
$$\left(\begin{array}{cc}0 & 0 \\ 0 & 1 \end{array}\right)$$
form a basis for the vector space of all 2 by 2 matrices. What is true for the basis is true for all 2 by 2 matrices.

 

[vdgreat]

but how can i prove it or generalize it??

 

[vdgreat]

help with this problem

anyone?

 

[morphism]

Did you think about what has been posted already?

 

[HallsofIvy]

Which of the basis matrices I gave commute with all other matrices?

 

[Marco_84]

try to look over schur lemma... it is a generalztion of what you asked...

ciao
marco

 

[vdgreat]

for each of the above matrices, i found out that it is true. but how can i prove this without having knowledge of basis. i haven't don't it yet.

 

[vdgreat]

i got it thanks