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 :)

First of all, notice that you can choose the rows of $A$ individually and independently. This simplifies the problem to be solved.
Each row $a$ of $A$ is a solution to $ax=0$ (the product of a $1\times 3$ matrix with the given $3\times1$ matrix, or a dot product if you prefer). This is a system of $1$ homogeneous linear equation in $3$ unknowns, so it will have a $2$-dimensional set of solutions (assuming $x$ is itself not the zero vector), which you can find by Gaussian elimination or even easier techniques. From the solution set you can choose any specific solutions you want for each row of $A$.
Moral: whenever a linear algebra problem can be translated into finding solutions to a system of linear equations, do it and go straight for Gaussian elimination!
