How to get the reminder of a vector divided by matrices

I try to get the reminder but not sure how to do so. Generally, f(z) = A(z)*fo(z) + r(z), f(z), fo(z), and r(z)are vertical 1 by n vectors of polynomials contain z, A are n by n matrices of polynomials contain z.The reminder r(z)needs to have a lower order than A(z). If only f and A are known, how to find r? for example, [z^2+1; 1]=[z 0; 0 1]*[z 1]+[1 0], f=[z^2+1; 1], A=[z 0; 0 1], fo=[z 1], r=[1 0].