Let $A$ be a $3 \times 3$ symmetric matrix with three distinct eigenvalues $1, -1$ and $\lambda$. Find $\lambda$ and the corresponding eigenvector.

Question

Let $A$ be a $3 \times 3$ symmetric matrix with three distinct eigenvalues $1, -1$ and $\lambda$. We know that $Ax = 0$ where

$$x =\begin{bmatrix} 0 \\ 1 \\ -1 \end{bmatrix}$$

I try to find $\lambda$ and the corresponding eigenvector. Do you have any ideas how to approach this problem?

Answer 1

If $Ax=0$ then $Ax=0×x$. So by definition of eigenvalue and eigenvector of a matrix $A$, it follows that $\lambda=0$ is an eigenvalue of $A$ [because there is a vector namely $x$ such that $Ax=\lambda x=0×x$], and $x$ as stated is an eigenvector of $A$ corresponding to the eigenvalue $\lambda=0$.