Understanding Cancellation in Matrix Multiplication: When Does AB AC Imply B C?

Matrix multiplication is a fundamental operation in linear algebra. However, unlike scalar multiplication, matrix multiplication is not always cancellative. Specifically, just because ( AB AC ) does not necessarily mean ( B C ). This article explores under what conditions the implication ( AB AC Rightarrow B C ) holds true and when it does not.

Conditions for Cancellation

The statement ( AB AC Rightarrow B C ) holds true only if the matrix ( A ) is invertible. This is because matrix multiplication by an invertible matrix can be thought of as a bijection (one-to-one and onto) that allows for cancellation.

Invertible Matrices

Let's start with the condition that ( A ) is invertible. If ( A ) is invertible, we can multiply both sides of the equation ( AB AC ) by the inverse of ( A ) from the left:

Multiply both sides by ( A^{-1} ): ( A^{-1}AB A^{-1}AC ). The left side simplifies to ( B ) and the right side simplifies to ( C ), yielding: ( B C ).

This shows that if ( A ) is invertible, then ( B C ).

Non-Invertible Matrices

However, if ( A ) is not invertible, the cancellation does not necessarily hold. One simple example is when ( A ) is the zero matrix. In this case, ( AB 0 ) and ( AC 0 ) for any matrices ( B ) and ( C ), regardless of whether ( B ) equals ( C ).

General Case and Nilpotent Matrices

More generally, if ( A ) is nilpotent with degree 2, we can choose ( B A ) and ( C 0 ) to disprove the statement ( AB AC Rightarrow B C ).

Ring Theoretic Perspective

The statement ( AB AC Rightarrow B C ) can be understood in the context of ring theory. Specifically, it is related to the concept of a right zero divisor and the trivial null space of the matrix ( A ).

Right Zero Divisors

A matrix ( A ) is called a right zero divisor if there exists a non-zero matrix ( B eq 0 ) such that ( AB 0 ). Conversely, ( A ) is not a right zero divisor if and only if ( AB 0 ) implies ( B 0 ).

Proof of Equivalence

Assume ( A ) satisfies right cancellation. Set ( C 0 ); then ( AB 0 ). Since ( B 0 ), ( A ) is not a zero divisor. Assume ( A ) is not a zero divisor. For ( AB AC ), we have ( AB - AC 0 ). Using distributive property, ( A(B - C) 0 ). Since ( A ) is not a zero divisor, ( B - C 0 ). Thus, ( B C ).

This result holds in any ring, not just matrices with entries in a field.

Special Case: Matrices with Entries in a Field

In the special case of matrices with entries in a field, being not a zero divisor is equivalent to being invertible. We can prove this:

If ( A ) is invertible, then ( AB 0 ) implies ( A^{-1}AB A^{-1}0 ), which simplifies to ( B 0 ). If ( A ) is a non-invertible matrix, the linear transformation ( x mapsto Ax ) is not injective. Therefore, there exists a vector ( v ) such that ( Av 0 ). Define matrix ( B ) to have columns equal to ( v ). Then ( AB [Av , Av cdots Av] 0 ). Hence, ( A ) is a zero divisor.

By taking the contrapositive, we conclude that not being a zero divisor implies invertibility.