Cancellation Property - Non-cancellative Algebras

Non-cancellative Algebras

Although, with the single exception of multiplication by zero and division of zero by another number, the cancellation law holds for addition, subtraction, multiplication and division of real and complex numbers, there are a number of algebras where the cancellation law is not valid.

The cross product of two vectors does not obey the cancellation law. If a×b = a×c, then it does not follow that b=c even if a≠0.

Matrix multiplication also does not necessarily obey the cancellation law. If AB=AC and A≠O, then one must show that matrix A is invertible (i.e. has det(A)≠0) before one can conclude that B=C. If det(A)=0, then B might not equal C, because the matrix equation AX=B will not have a unique solution for a non-invertible matrix A.