Inverting matrix multiplication "and" representing with a smaller sized matrix

Question

Consider I have a vector $A=[a_0 \ \ a_1]$ and a random binary matrix $B$ which is $2\times 2$. I compute $C=A\cdot B$.

My question is: " Can one compute $B$ Given $C$ and $A$? "

Note: By binary matrix I mean the entries of matrix $B$ can be $0$ or $1$. Also not all entries of $A$ and $B$ are zero. $A$ can be a binary vector too.

Answer 1

Let $$ B=\begin{bmatrix} b_1 & b_2 \\ b_3 & b_4 \end{bmatrix} $$ where the $b_i$ are all $0$ or $1$. Then $$ C=AB=\begin{bmatrix} a_1b_1+a_2b_3 & a_1b_2+a_2b_4 \end{bmatrix}. $$ Suppose that $A=\begin{bmatrix} 1 & 0 \end{bmatrix}$. Then $C=\begin{bmatrix} b_1 & b_2 \end{bmatrix}$. So, you would not be able to recover $b_3$ and $b_4$.

Answer 2

There's no straightforward way to compute B from C and A since C and A are nonsquare and therefore noninvertible.