[Math] Find a matrix from its eigenvalues and corresponding vectors

Question

Suppose $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda_1=-1$ $\lambda_2=0$ and $\lambda_3=1$ and with the corresponding eigenvectors $\vec{v_1}=<1,0,2>$ $\vec{v_2}=<-1,1,0>$ and $\vec{v_3}=<0,0,1>$

Find matrix the $A$

So I made $P=$
$\begin{bmatrix}
1 & -1 & 0 \\
0 & 1 & 0 \\
2 & 0 & 1 \\
\end{bmatrix}$
and got $P^{-1} =$
$\begin{bmatrix}
1 & 1 & 0 \\
0 & 1 & 0 \\
-2 & -2 & 1 \\
\end{bmatrix}$

I am unsure where to go from here though? I feel as though maybe there is missing information in the question?

Answer 1

The searched matrix is $M=PDP^{-1}$ where $D$ is the diagonal matrix that has as diagonal elements the eigenvalues, in the same order as the eigenvectors in $P$ (see here).

$$ M= \begin{bmatrix} 1&-1&0\\ 0&1&0\\ 2&0&1 \end{bmatrix} \begin{bmatrix} -1&0&0\\ 0&0&0\\ 0&0&1 \end{bmatrix} \begin{bmatrix} 1&1&0\\ 0&1&1\\ -2&-2&1 \end{bmatrix} = \begin{bmatrix} -1&-1&0\\ 0&0&0\\ -4&-4&1 \end{bmatrix} $$