# Determine the matrix if we know eigenvalues and eigenvectors

I have been doing one quite simple task and there is one thing that I do not understand.

I am aware of the following regarding eigenvectors: If A is an n×n matrix, the nonzero n-component column vector x is an eigenvector for eigenvalue λ if Ax=λx.

Here is the exercise: The matrix A has the eigenvalues −1 and 2 with corresponding eigenvectors $\begin{bmatrix}0 \\ 1\end{bmatrix}$ and $\begin{bmatrix} 2 \\ 1 \end{bmatrix}$.

Compute A.

Now: Let A be $$\begin{bmatrix} a & b\\ c & d\\ \end{bmatrix}$$ It follows that:

My question is: Given the statement at the top of this post should not the following apply instead?: $A \cdot \begin{bmatrix}2\\1\end{bmatrix}$ = $2 \cdot \begin{bmatrix}2\\1\end{bmatrix}$

• Yes, it's a typo – Argyll Mar 24 '18 at 1:10

It is a printing mistake. In particular, if $\begin{pmatrix} 2 \\ 1\end{pmatrix}$ is an eigenvector of $A$ with eigenvalue $2$, then: $$A\begin{pmatrix}2 \\ 1\end{pmatrix} = 2 \require{enclose}\enclose{updiagonalstrike,downdiagonalstrike}{\begin{pmatrix} 0 \\ 1\end{pmatrix}} {\begin{pmatrix}2 \\ 1\end{pmatrix}}$$ by the definition of eigenvector. You can make the correction and proceed.