Commuting Matrices

Commuting matrices are two matrices and that satisfy

(1)

under matrix multiplication.

In general, matrix multiplication is not commutative. Furthermore, a nonzero square matrix need not have a matrix inverse: exists exactly when , where is the determinant. Finally, can be zero even without or . Moreover, when , we may still have , a simple example of which is provided by

			(2)
			(3)

for which

(4)

but

(5)

(Taussky 1957).

For a noncommutative matrix ring, its commuting graph records commuting pairs outside its ring center.