# Find all matrices {(a,b);(c,d)} that will commute with every matrix in set S

• Aug 15th 2010, 11:49 AM
experiment00
Find all matrices {(a,b);(c,d)} that will commute with every matrix in set S
where set S is the set of all 2x2 matrices

so i set an arbitrary matrix from the set S {(w,x);(y,z)}

so (sorry i dont know the matrix format :< )

[a b][w x] = [w x][a b]
[c d][y z] [y z] [c d]

and got

[(ax+bz) (ay+bw)] = [(xa+yc) (xb+yd)]
[(cx+dz) (cy+dw)] = [(za+wc) (zb+wd)]

and then equating them but im stuck here, lol xD
seems like i forgot algebra! XD any help please?
• Aug 15th 2010, 12:07 PM
NonCommAlg
so your matrix must commute with $e_{11}=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$ and $e_{12}=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}.$ thus ...
• Aug 15th 2010, 12:09 PM
experiment00
hmm, why should they commute with that 2 specific matrices sir? :( sorry
• Aug 15th 2010, 03:23 PM
Vlasev
Well, first you want it to commute with all matrices so you might as well chose some. The more important reason is that it is easy to work with these easy matrices. If you use all four of these matrices with one entry 1, you can build all other matrices as a linear combination of those four. If your matrix commutes with each of the four, then it should commute with all linear combinations of these four.