Editing Matrix multiplication (section)

===Product with a scalar===
If {{math|'''A'''}} is a matrix and {{mvar|c}} a scalar, then the matrices <math>c\mathbf{A}</math> and <math>\mathbf{A}c</math> are obtained by left or right multiplying all entries of {{math|'''A'''}} by {{mvar|c}}. If the scalars have the [[commutative property]], then <math>c\mathbf{A} = \mathbf{A}c.</math>

If the product <math>\mathbf{AB}</math> is defined (that is, the number of columns of {{math|'''A'''}} equals the number of rows of {{math|'''B'''}}), then 
:<math> c(\mathbf{AB}) = (c \mathbf{A})\mathbf{B}</math> and <math> (\mathbf{A} \mathbf{B})c=\mathbf{A}(\mathbf{B}c).</math>
If the scalars have the commutative property, then all four matrices are equal. More generally, all four are equal if {{math|''c''}} belongs to the [[Center (ring theory)|center]] of a [[ring (mathematics)|ring]] containing the entries of the matrices, because in this case, {{math|''c'''''X''' {{=}} '''X'''''c''}} for all matrices {{math|'''X'''}}.

These properties result from the [[bilinearity]] of the product of scalars:
:<math>c \left(\sum_k a_{ik}b_{kj}\right) = \sum_k (c a_{ik} ) b_{kj} </math>
:<math>\left(\sum_k a_{ik}b_{kj}\right) c = \sum_k a_{ik} ( b_{kj}c). </math>