Show the matrix $~P~$ of the eigenvectors that have the corresponding eigenvalues $~λ_1,~ λ_2,~ λ_3~$

Question

I am trying to solve matrix practice examples for my upcoming exam. I am kinda stuck on this question and don't really know what to do, any help would be appreciated

Consider a matrix $~A~$,

\begin{bmatrix}0&1&0\\0&0&1\\-a_0&-a_1&-a_2\end{bmatrix}

Assume that the corresponding eigenvalues are $~λ_1,~ λ_2,~ λ_3~$. Show that the matrix $~P~$ of the eigenvectors is

\begin{bmatrix}1&1&1\\λ_1&λ_2&λ_3\\λ_1&λ_2&λ_3\end{bmatrix}

Answer 1

Hints:

• There's a typo in your matrix of eigenvectors. It should be $$ \pmatrix{1&1&1\\ \lambda_1&\lambda_2&\lambda_3\\ \lambda_1^2&\lambda_2^2&\lambda_3^2\\ } $$
• The characteristic polynomial of the matrix $\ A\ $ is $\ x^3+a_2x^2+a_1x+a_0\ $, so each of the eigenvalues $\ \lambda_i\ $ satisfies the equation $\ -a_0-a_1\lambda_i-a_2\lambda_i^2=\lambda_i^3\ $