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 :)

Given that we don't really know anything about A (other than non-zero) we can find values to satisfy the equation, but it will not be unique at all. This question breaks down into a system of three equations, but with 9 unknowns. 
$$
\begin{pmatrix}
  a & b & c \\
  d & e & f \\
  g & h & i
 \end{pmatrix}
\begin{pmatrix}
  x \\
  y \\
  z
 \end{pmatrix}
=
\begin{pmatrix}
  0 \\
  0 \\
  0
 \end{pmatrix}
$$
So it comes down to selecting values that satisfy the three corresponding equations. 
$$
 \left\{
  \begin{array}{l l}
    ax+by+cz=0\\
    dx+ey+fz=0 \\
    gx+hy+iz=0
  \end{array} \right\}.
$$
Again, these values wont be unique, but in this way you can get a matrix A to satisfy your equation.
