Matrix AB=AC, A is not a zero matrix, so it cannot be inferred that B and C are equal. Why? When a is m*n matrix and r(A)=n, the proposition holds. Why?

There are so many people asking about this deformation. . . .

AB=AC, then A(B-C)=0.

So B-C is a matrix composed of vectors in the solution space of Ax=0.

Even if A is not a zero matrix, Ax=0 can have a non-zero solution, so B-C is not necessarily equal to zero.

And a is an m*n matrix. When r(A)=n, Ax=0 can only have zero solution, so B-C=0, so B = C.