Eigenvalues & eigenvectors of a matrix

Question

I have a couple of questions regarding eigenvalues and eigenvectors.

Let $A=\begin{pmatrix}4 & 2 \\ 5 & 1\end{pmatrix}$, $\mathbf{u}=\begin{pmatrix}2\\-5\end{pmatrix},\mathbf{v}=-2\mathbf{u}$ and $\mathbf{w}=\begin{pmatrix}1\\-1\end{pmatrix}$

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1. How would I be able to show that u is an eigenvector of A as well as finding the corresponding eigenvalue?

2. How would I be able to tell if v is an eigenvector of A? And w is an eigenvector of A?

Answer 1

Hint

Keep in mind the definition of an eigenvector:

• $v$ is an eigenvector of the matrix $A$ if $Av=\lambda v$, for some $\lambda\in\mathbb{R}$.

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Question 1

$Av=\begin{pmatrix}4&2\\5&1\end{pmatrix}\begin{pmatrix}2\\-5\end{pmatrix}=\begin{pmatrix}4\cdot2+2\cdot-5\\5\cdot2+1\cdot-5\end{pmatrix}=\begin{pmatrix}-2\\0\end{pmatrix}$

Since there is no $\lambda$ for which $Av=\lambda v$, $v$ is not an eigenvector of $A$.