Find a minimal polynomial of some matrix given a minimal polynomial of another matrix.

Question

I would like an approach on how to approach questions where I'm given a minimal polynomial $m_B(\lambda)$ (minimal polynomial of matrix B) and I'm asked to find the minimal polynomial of the matrix $2(B-I)$.

For instance, given $m_B(\lambda) = x(x^2+3)$ find $m_{2B-2I}(\lambda)$.

I tried to find $det(2B-2I)$ but that leads to nowhere. What I'm thinking of is maybe using the fact that $Cx = \alpha x$. However, I'm not sure whether these 2 matrices have the same eigenvalues or not.

Answer 1

HINT

Let $M=2(B-I)$ and then $B=I+\frac{1}{2}M$. Substitute this for $B$ in the given minimal polynomial.

You now have a polynomial satisfied by $M$ and all that remains is to check that no smaller polynomial is possible.