# Showing eigenvalue belongs to a matrix and basis of eigenspace

The question I am trying to answer is: Show that 4 is an eigenvalue of B and give the corresponding basis of the eigenspace.

The following matrix B and vector v are given:

B= $\begin{pmatrix} 2 & 2 & 2\\ 1 & 3 & -1\\ 1 & -1 & 3 \end{pmatrix}$.

v= $\begin{pmatrix} -2\\ 1\\ 1 \end{pmatrix}$.

Now I know how to show that v is an eigenvector of B and how to get it's corresponding eigenvalue, but I'm stuck on how to show that a certain value is an eigenvalue. Could someone show me how one would go about doing this? And how do I get the corresponding basis of the eigenspace from that?

Thanks!

You can show a certain value is an eigenvalue either by finding its corresponding eigenvector, or by checking that $\det(\lambda I - B) = 0$, where $\lambda$ is the potential eigenvalue. The eigenspace of $\lambda$ is the set of vectors $v$ such that $Bv = \lambda v$, once you've found an expression for the eigenvectors that should become clear to you.