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I am confused of some commutative properties of some matrices, so here is the question.

What would constitute(or be the name of) a matrix that is always commutative? Which matrices would satisfy this property?

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    The term commutative refers to an operation, not to the objects. Two objects can commute. One object can neither commute nor be commutative.2012-09-27

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Hint: try to prove that such a matrix have the form $\lambda I$ where $\lambda\in\mathbb{F}$ by following these steps:

  1. Show that the matrix can't have a non-zero coordinate that is not on the diagonal

  2. Show that all the coordinate on the diagonal must be the same

To do both steps assume by negation and construct a matrix that does not commute with your matrix to get a contradiction.

The name of all such matrices is scalar matrices

Note: this is an answer for the case of square matrices

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Unit matrices $\bf 1$ commute ($[A,B]_-=AB-BA=0$) with all other matrices of the same dimension.

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    This answer is wrong. The only matrices that always commute with any other matrix (of the same square order, of course) are the scalar ones, as pointed out in Belgi's answer.2012-09-27