# How to find the kernel of a matrix

So I did read some other answers, but I'm not sure if what I ve done is right and how to conclude. ( find the basis of the kernel of matrix A )

I'm asked to find a basis of the kernel of this matrix :

$$A= \begin{pmatrix} 2 & 5 & -1 & 2\\ -2 & -16 & -4 & 4 \end{pmatrix}$$

I ve found this :

$$\begin{pmatrix} 6x \\ y \\ z \\ 6t \end{pmatrix} = y \begin{pmatrix} -26 \\ 1 \\ 0 \\ 11 \end{pmatrix} + z \begin{pmatrix}-12 \\ 0 \\ 1 \\ 5 \end{pmatrix}$$

Do you have a fast method for doing a problem like this ? And how do you conclude from this point?

I need to precise something, I'm working on $$\mathbb Z$$. My ultimate goal is to find the structure of $$\mathbb Z^{(3)} / K$$ where $$K$$ is the kernel.

• – amd Jan 12 at 21:30
• this situation does not apply to my case because in the diagonal of my matrix, i dont necessarly have only 1, because we are working in Z not in R – Marine Galantin Jan 14 at 15:19
• The method can be adapted without much grief. – amd Jan 14 at 19:11