# Thread: Showing that vectors are eigenvectors of a matrix

1. ## Showing that vectors are eigenvectors of a matrix

The question:

Show that (1, -2) and (2, -3) are eigenvectors of A and find the corresponding eigenvalues, for

$\displaystyle $A = \left( {\begin{array}{cc} -6 & -4 \\ 12 & 8 \\ \end{array} } \right)$$

My attempt:

I just multipled the eigenvectors by the matrix, then found the eigenvalues by inspecting which multiple to use to get this new vector. This gets the correct solution, but I'm wondering if this is sufficient since it says to "show that" they're eigenvectors.

Thanks.

2. In order to show that x is an eigenvector of A, all you have to do is show that the equation $\displaystyle Ax=\lambda x$ holds for your nonzero candidate $\displaystyle x.$ So, if $\displaystyle A$ acts on $\displaystyle x$ merely by changing its magnitude, and not its direction, then you're done. Your method of proof is perfectly adequate for this purpose.

3. Thank you very much.

4. You're welcome!