# Thread: Finding a value in a matrix with distinct eigenvalues

1. ## Finding a value in a matrix with distinct eigenvalues

The matrix $\displaystyle A=\left(\begin{array}{ccc}-1&1&0\\-6&6&1\\k&0&0\end{array}\right)$

has three distinct real eigenvalues if and only if $\displaystyle x<k<y$. Find $\displaystyle x$ and $\displaystyle y$.

2. Originally Posted by chewitard
The matrix $\displaystyle A=\left(\begin{array}{ccc}-1&1&0\\-6&6&1\\k&0&0\end{array}\right)$

has three distinct real eigenvalues if and only if $\displaystyle x<k<y$. Find $\displaystyle x$ and $\displaystyle y$.

1) Calculate the characteristic polynomial of A ;

2) A polynomial $\displaystyle f(x)$ has a multiple root $\displaystyle \alpha$ iff $\displaystyle f(\alpha)=f'(\alpha)=0$ ;

3) Using the above check what could be be the possible multiple roots of the char. pol. of A and what have to be the values of k that'd allow such a thing to happen.

Tonio