Question

Find a 2 × 2 nondiagonal matrix whose eigenvalues are 5 and −3, and associated eigenvectors are −1 2 and 1 1 , respectively.

Answer 1

You're given the diagonal decomposition of the matrix:

$\begin{bmatrix}-1&1\\2&1\end{bmatrix}\begin{bmatrix}5&0\\0&-3\end{bmatrix}\begin{bmatrix}-1&1\\2&1\end{bmatrix}^{-1}$

Computing the product yields

$-\dfrac13\begin{bmatrix}1&8\\16&7\end{bmatrix}$