# Commutation property

## Main Question or Discussion Point

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 dont 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..

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morphism
Homework Helper
Try particular As, like

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

and see where that leads you.

HallsofIvy
Homework Helper
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.

but how can i prove it or generalize it??

help with this problem

anyone?

morphism
Homework Helper

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

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

ciao
marco

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 dont it yet.

i got it thanks