Linear algebra eigenvalues and eigenvectors

Question

Suppose that $p(\lambda) = \lambda^3 -7\lambda + 6$ is a characteristic polynomial of the matrix $A$. Find the eigenvalues of $A$.
Is it possible to determine the order of the square matrix $A$?
Hint: if $r$ is a root of $p(\lambda)$, then $\lambda - r$ is a factor of $p(\lambda)$.

Answer 1

Hint 1: if $p(\lambda)=\lambda^3-7\lambda+6$, then $p(1)=0$, so the polynomial is divisible by $\lambda-1$; can you find $q(\lambda)$ such that $$ p(\lambda)=(\lambda-1)q(\lambda) $$ and tell what's the degree of $q$? Can you find the roots of $q$?

Hint 2: the order of a matrix and the degree of its characteristic polynomial are…