# Finding matrix from plane in kernal

Give an example of a matrix A such that $$\ker(A)$$ is the plane $$2x − y + 3z = 0$$.

I am not sure where to start, as I know that the $$\ker(A)$$ is the matrix of the plane, but I don't know how to go backwards from it. Would I need to assume that there's a free variable in the rref(A) and assume it was pulled out of the equation of the plane?

Let $$A=\begin{pmatrix} 2 & -1 & 3\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{pmatrix}$$
Then $$A\begin{pmatrix} x\\ y\\ z\end{pmatrix}=0$$ implies $$2x-y+3z=0$$
• @amd You are right. I am just used to considering linear transformations in $n$-dimensional vector space $V$: $V\to V$, so my matrix is a square matrix. – Bach May 21 at 3:55