how to find a matrix A given the solution? if we need,for example, to find a nonzero 3x3 matrix A such that we are given a 3x1 vector as a solution to Ax = 0. What is the general procedure we can follow to obtain such Matrix A?
Thank you :)

Let's suppose that your vector $v$ is a column vector. 
One of options is to look at the matrix $B:=vv^T$: it's a $3\times 3$ matrix, and $Av = \|v\|^2v$. Now we can look at the matrix $A:=(\|v\|^2Id-B)$: easy to check that $v$ belongs to its nullspace. We need to check that our $A$ is not zero; indeed, take any nonzero vector $w$ such that $v\bot w$, then $Aw=\|v\|^2w\ne 0$.
Note that $A$ is not uniquely defined, because you have $3$ linear equations on $9$ elements of the matrix.
