## Linear Algebra: A Modern Introduction

$\lambda=-1$
If A is an nxn matrix, a scalar $\lambda$ is called an eigenvalue of A is there is a non-zero vector $v$ such that $Ax=\lambda x$. This vector is called the eigenvector of A corresponding to $\lambda$. Hence here, we compute: $Av=\begin{bmatrix} 1 &2 \\ 2 & 1 \end{bmatrix}\cdot\begin{bmatrix} 3 \\ -3 \end{bmatrix}=\begin{bmatrix} -3\\ 3 \end{bmatrix}=-1\begin{bmatrix} 3 \\ -3 \end{bmatrix}=-1v$ Thus we can see that $\lambda=-1$