Solving system of equations Mathematica

[Mr Davis 97]

I am solving the following system of equations (in matrix form):

$\begin{pmatrix} 1 & -2 & -1 & 1 \\ 2 & -3 & 1 & 6 \\ 3 & -5 & 0 & 7 \\ 1 & 0 & 5 & 9 \end{pmatrix}$

I want to solve it using Mathematica, but when I use the command LinearSolve[], I only get back $\begin{pmatrix} 9 \\ 4 \\ 0 \end{pmatrix}$, which is a particular solution. However, I am looking for how to get an output of something like $\begin{pmatrix} 9 \\ 4 \\ 0 \end{pmatrix} + t_1\begin{pmatrix} -5 \\ -3 \\ 1 \end{pmatrix}$, where the linear combinations of the basis for the kernel is included. Is there a single command that does this? Or would I have to separately the NullSpace[] command?

 

[Buzz Bloom]

Hi Mr Davis:

I know next to nothing about Mathematica, but I would expect it to have a function you could invoke that would invert a matrix for you. Did you look for one to do that?

ADDED
Sorry, I misunderstood your question. Since you got a 3d vector as the solution from a 4d matrix, I am guessing the determinant is zero. I don't understand why the answer would be a 3d vector, so I am at a loss. I would expect, and I am guessing you are looking for, an answer of the form:

V1 + const x V2​

where V1 and V2 are 4d vectors. V1 is a solution to

M x V = V3,​

and V2 is a solution for

M x V = 0.​

I have nothing to suggest about how to approach this using Mathematica.

Regards,
Buzz