# Suppose a 2 × 2 matrix A has an eigenvector (1 2) , with corresponding eigenvalue −4. what is A(-2 -4) ?

May 4, 2017

$A \left(\left(- 2 , - 4\right)\right) = \left(\left(8 , 16\right)\right)$

#### Explanation:

If $l a m \mathrm{da}$ is an eigenvalue with corresponding eigenvector $\underline{v}$ then:

$A \underline{v} = l a m \mathrm{da} \underline{v}$

From which we get with $l a m \mathrm{da} = - 4$ and $\underline{v} = \left(\left(1 , 2\right)\right)$ :

$A \left(\left(1 , 2\right)\right) = - 4 \left(\left(1 , 2\right)\right) \setminus \setminus \setminus \ldots . . \left(\star\right)$

A properties of matrices is that $A \mu B = \mu A B$, and so

$A \left(\left(- 2 , - 4\right)\right) = \left(- 2\right) A \left(\left(1 , 2\right)\right)$
$\text{ } = \left(- 2\right) \left(- 4\right) \left(\left(1 , 2\right)\right)$ using $\left(\star\right)$
$\text{ } = 8 \left(\left(1 , 2\right)\right)$
$\text{ } = \left(\left(8 , 16\right)\right)$

May 4, 2017

By definition, the eigenvectors ($m a t h b f {e}_{i}$) of $A$, and its eigenvalues (${\lambda}_{i}$) are related as:

$A m a t h b f {e}_{i} = {\lambda}_{i} m a t h b f {e}_{i}$

Suppose a 2 × 2 matrix A has an eigenvector (1 2) , with corresponding eigenvalue −4. what is A(-2 -4) ?

This means that:

$A \left(\begin{matrix}1 \\ 2\end{matrix}\right) = - 4 \left(\begin{matrix}1 \\ 2\end{matrix}\right)$

So with the question what is A(-2 -4) ?:

$- 2 \left(A \left(\begin{matrix}1 \\ 2\end{matrix}\right)\right) = - 2 \left(- 4 \left(\begin{matrix}1 \\ 2\end{matrix}\right)\right)$

$\implies A \left(\begin{matrix}- 2 \\ - 4\end{matrix}\right) = \left(\begin{matrix}8 \\ 16\end{matrix}\right)$