Do matrices form an abelian group under multiplication?

Answer 1

More precisely, I think you're asking whether the set of n X n matrices forms an abelian group under multiplication. The answer is no (assuming n>1). For example

(1 0)(0 1) = (0 1)

(0 0)(0 0) (0 0),

but

(0 1)(1 0) = (0 0)

(0 0)(0 0) (0 0). However, the set of n x n diagonalmatrices does form an Abelian set. This is true regardless of the direction of the diagonality, right-to-left or left-to-right. Note that the resulting matrix will also be diagonal, but always right-to-left.