Editing Cancellation property (section)

== Non-cancellative algebraic structures ==
Although the cancellation law holds for addition, subtraction, multiplication and division of [[real number|real]] and [[complex number]]s (with the single exception of multiplication by [[0 (number)|zero]] and division of zero by another number), there are a number of algebraic structures where the cancellation law is not valid.

<!-- The [[vector (spatial)|vector]] [[dot product]] is perhaps the simplest example. In this case, for an arbitrary nonzero vector '''a''', the product {{nowrap|1='''a''' ⋅ '''b'''}} can equal another dot product {{nowrap|1='''a''' ⋅ '''c'''}} even if {{nowrap|'''b''' ≠ '''c'''}}. This occurs because the dot product relates to the angle between two vectors as well as their magnitude, and a change in one can, in effect, counterbalance the other to produce equal products for unequal vectors.

For the same reason, the-->The [[cross product]] of two vectors <!--also--> does not obey the cancellation law. If {{nowrap|1='''a''' × '''b''' = '''a''' × '''c'''}}, then it does not follow that {{nowrap|1='''b''' = '''c'''}} even if {{nowrap|'''a''' ≠ '''0'''}} (take {{nowrap|1='''c''' = '''b''' + '''a'''}} for example)

<!-- However, if ''both'' '''a''' · '''b'''='''a''' · '''c''' ''and'' '''a''' × '''b''' = '''a''' × '''c''', then one ''can'' conclude that '''b''' = '''c'''. This is because for dot and cross products to be simultaneously equal, then both '''a''' · ('''b''' - '''c''') ''and'' '''a''' × ('''b''' - '''c''') must be zero by the [[distributive law]]. This means that both the sine and cosine of the angle between '''a''' and ('''b''' - '''c''') must be zero, which is not possible because sin<sup>2</sup>''x'' + cos<sup>2</sup>''x'' is ''identically'' 1.-->
[[Matrix multiplication]] also does not necessarily obey the cancellation law. If {{nowrap|1='''AB''' = '''AC'''}} and {{nowrap|'''A''' ≠ 0}}, then one must show that matrix '''A''' is ''invertible'' (i.e. has {{nowrap|[[determinant|det]]('''A''') ≠ 0}}) before one can conclude that {{nowrap|1='''B''' = '''C'''}}. If {{nowrap|1=det('''A''') = 0}}, then '''B''' might not equal '''C''', because the [[matrix (mathematics)|matrix]] equation {{nowrap|1='''AX''' = '''B'''}} will not have a unique solution for a non-invertible matrix '''A'''.

Also note that if {{nowrap|1='''AB''' = '''CA'''}} and {{nowrap|'''A''' ≠ 0}} and the matrix '''A''' is ''invertible'' (i.e. has {{nowrap|[[determinant|det]]('''A''') ≠ 0}}), it is not necessarily true that {{nowrap|1='''B''' = '''C'''}}. Cancellation works only for {{nowrap|1='''AB''' = '''AC'''}} and {{nowrap|1='''BA''' = '''CA'''}} (provided that matrix '''A''' is ''invertible'') and not for {{nowrap|1='''AB''' = '''CA'''}} and {{nowrap|1='''BA''' = '''AC'''}}.