If we have a matrix multiplication equation,
$$s=A^\top \cdot P \cdot B,$$
Where $s$ is an integer and I have to find the matrix $P$. How would I use matrix alegbra here, since $A$ and $B$ are vectors and therefore the inverse cannot be found.
Thanks :)
Assume $B$ is nonzero (otherwise $s=0$). So let $B=(b_1,....,b_n)$ and assume $b_i\neq 0$ for some $i$. Then the following choice for $P$ satisfies $A^TPB=s$: Fill $P$ with zeros, but in the $i$-th column, insert $\frac{s}{b_i\lvert|A\rvert|^2}A$. Then $PB=\frac{s}{\lvert|A\rvert|^2}A$. (if $\lvert|A\rvert|=0$, this does not work, but then $A=0$ and $s=0$.)