Solve the following system of equations by using the inverse of the coefficient matrix A. (AX=B), x+5y=-10, -2x+7y=-31

For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A.
$A=\left[\begin{array}{cccc}2& 3& 5& -9\\ -8& -9& -11& 21\\ 4& -3& -17& 27\end{array}\right]$
Find a nonzero vector in Nul A.
$A=\left[\begin{array}{c}-3\\ 2\\ 0\\ 1\end{array}\right]$

Find an explicit description of Nul A by listing vectors that span the null space.
$A=\left[\begin{array}{ccccc}1& 5& -4& -3& 1\\ 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

Assume that A is row equivalent to B. Find bases for Nul A and Col A.
$A=\left[\begin{array}{ccccc}1& 2& -5& 11& -3\\ 2& 4& -5& 15& 2\\ 1& 2& 0& 4& 5\\ 3& 6& -5& 19& -2\end{array}\right]$
$B=\left[\begin{array}{ccccc}1& 2& 0& 4& 5\\ 0& 0& 5& -7& 8\\ 0& 0& 0& 0& -9\\ 0& 0& 0& 0& 0\end{array}\right]$

If a nonzero matrix $A$ is transformed from ${\mathbb{R}}^3$ to ${\mathbb{R}}^2$, then the null space of $A$ must be a one dimensional (sub)space of ${\mathbb{R}}^3$.
So i know that null space of $A$ is $\{x:Ax=0\}$ and I also know the definition of not onto. I don't understand the whole concept of one-dimensional space and would this statement be always true?

The given matrix is the augmented matrix for a system of linear equations. Give the vector form for the general solution.
${\displaystyle \left[\begin{array}{cccccc}1& -1& 0& -2& 0& 0\\ 0& 0& 1& 2& 0& 0\\ 0& 0& 0& 0& 1& 0\end{array}\right]}$

If A is an n x n matrix , where are the entries on the main diagonal of A-A^T? Justify yoyr answer.

The problem is this:
The impulse response of a system is the output from this system when excited by an input signal $\delta (k)$ that is zero everywhere, except at $k=0$, where it is equal to 1. Using this definition and the general form of the solution of a difference equation, write the output of a linear system described by:
$y(k)–3y(k–1)–4y(k–2)=\delta (k)+2\delta (k–1)$
The initial conditions are: $y(–2)=y(–1)=0$.
My question is: How can the particular solution be found using the method of undetermined coefficients if the non-homogeneous equation is also a difference equation?