Find a Matrix A on the ring of integers modulo 3 so that KerA=ImB.

Question

B={{1,1,1},{0,1,2},{2,1,0},{0,2,2}}

I understand that each vector from then span of column vectors of B is a solution for Ax=o and that matrix A should have four columns. However I don't know how many rows it should have and how to find it.

Answer 1

Suppose $A$ has $r$ rows. As $\mathbb Z_3$ is a field, by rank-nullity theorem, $$ 4 =\text{dimension of domain of $A$}=\operatorname{rank}(A)+\operatorname{nullity}(A)=\operatorname{rank}(A)+\operatorname{rank}(B). $$ Hence $r:=\operatorname{rank}(A)=4-\operatorname{rank}(B)$ and $A$ must have at least $r$ rows.