<s>[INST] Find matrix from Eigenvectors and Eigenvalues A matrix $A$ has eigenvectors
$v_1 = \left(  
\begin{array}{c}
    2 \\
    1 \\
  \end{array}
\right)$ 
$v_2 = \left(  
\begin{array}{c}
    1 \\
    -1 \\
  \end{array}
\right)$
with corresponding eigenvalues $\lambda_1$= 2 and $\lambda_2$= -3, respectively.
Determine Ab for the vector b = $ 
\left(  
\begin{array}{c}
    1 \\
    1 \\
  \end{array}
\right)$
I know how to find eigenvalues and eigenvectors from a given matrix A, but not this one,
the vector A is a 2x1 matrix, determinant does not exist here, so how to find the matrix A as stated in the question?
 [/INST] By definition of eigenvalue and eigenvector, we have 
$$\tag{1}A\left(  
\begin{array}{c}
    2 \\
    1 \\
  \end{array}
\right)=2\left(  
\begin{array}{c}
    2 \\
    1 \\
  \end{array}
\right)\mbox{ and }A\left(  
\begin{array}{c}
    1 \\
    -1 \\
  \end{array}
\right)=-3\left(  
\begin{array}{c}
    1 \\
    -1 \\
  \end{array}
\right).$$
Now, since 
$$\left(  
\begin{array}{c}
    1 \\
    1 \\
  \end{array}
\right)=\frac{2}{3}\left(  
\begin{array}{c}
    2 \\
    1 \\
  \end{array}
\right)-\frac{1}{3}\left(  
\begin{array}{c}
    1 \\
    -1 \\
  \end{array}
\right),$$
we have
$$A\left(  
\begin{array}{c}
    1 \\
    1 \\
  \end{array}
\right)=\frac{2}{3}A\left(  
\begin{array}{c}
    2 \\
    1 \\
  \end{array}
\right)-\frac{1}{3}A\left(  
\begin{array}{c}
    1 \\
    -1 \\
  \end{array}
\right)=....\mbox{(using $(1)$)}$$
</s>