Editing Matrix multiplication (section)

===Non-commutativity===
An operation is [[commutative property|commutative]] if, given two elements {{math|'''A'''}} and {{math|'''B'''}} such that the product <math>\mathbf{A}\mathbf{B}</math> is defined, then <math>\mathbf{B}\mathbf{A}</math> is also defined, and <math>\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}.</math>

If {{math|'''A'''}} and {{math|'''B'''}} are matrices of respective sizes {{tmath|m\times n}} and {{tmath|p\times q}}, then <math>\mathbf{A}\mathbf{B}</math> is defined if {{tmath|1=n=p}}, and <math>\mathbf{B}\mathbf{A}</math> is defined if {{tmath|1=m=q}}. Therefore, if one of the products is defined, the other one need not be defined. If {{tmath|1=m=q\neq n=p}}, the two products are defined, but have different sizes; thus they cannot be equal.  Only if {{tmath|1=m=q= n=p}}, that is, if {{math|'''A'''}} and {{math|'''B'''}} are [[square matrices]] of the same size, are both products defined and of the same size. Even in this case, one has in general
:<math>\mathbf{A}\mathbf{B} \neq \mathbf{B}\mathbf{A}.</math> 
For example
:<math>\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}=\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},</math>
but
:<math>\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}.</math>

This example may be expanded for showing that, if {{math|'''A'''}} is a {{tmath|n\times n}} matrix with entries in a [[field (mathematics)|field]] {{mvar|F}}, then <math>\mathbf{A}\mathbf{B} = \mathbf{B}\mathbf{A}</math> for every {{tmath|n\times n}} matrix {{math|'''B'''}} with entries in {{mvar|F}}, [[if and only if]] <math>\mathbf{A}=c\,\mathbf{I}</math> where {{tmath|c\in F}}, and {{math|'''I'''}} is the {{tmath|n\times n}} [[identity matrix]]. If, instead of a field, the entries are supposed to belong to a [[ring (mathematics)|ring]], then one must add the condition that {{mvar|c}} belongs to the [[center (ring theory)|center]] of the ring.

One special case where commutativity does occur is when {{math|'''D'''}} and {{math|'''E'''}} are two (square) [[diagonal matrices]] (of the same size); then {{math|1='''DE''' = '''ED'''}}.<ref name=":2" /> Again, if the matrices are over a general ring rather than a field, the corresponding entries in each must also commute with each other for this to hold.