Identity Matrix: Does AB=BA for Other Matrices?

[EvLer]

Is the identity matrix (or multiple of) the only one that commutes with other matrices or are there other matrices that AB=BA?

Thanks

 

[StatusX]

Yes, those are the only ones that commute with all other matrices. There are pairs of matrices which aren't multiples of the identity matrix that commute with each other, like any matrix with a multiple or power of itself, or:

$$\left[ \begin{array}{cc}0 & 1 \\ 1 & 0 \end{array} \right], \left[ \begin{array}{cc}a & b \\ b & a \end{array} \right]$$

but they won't commute with an arbitrary matrix.

 

[loom91]

The identity matrix commutes only when both IA and AI are defined.